Lifting foil

ABSTRACT

A lifting foil having a lower trailing course having a margins and an upper leading course margins connected in a manner to enhance lift of the foil.

This application claims the benefit of earlier filed provisionalapplication 61/085,904 filed Aug. 4, 2008.

BACKGROUND OF THE INVENTION

This invention relates generally to the field of fluid dynamics withemphasis on aerodynamic drag phenomena. A particular object of theinvention is to reduce energy losses suffered by an aircraft in flightdue to induced drag. It is known that this particular type of drag isaccompanied by a shedding of vortices from the tips of the wings.

The present inventor discussed the details of drag issues surroundingwing tips in commonly owned U.S. Pat. No. 7,100,867 to Houck II. In thatpatent it was explained that these vortices result from a span-wise flowof air from a relatively high pressure condition on the lower wingsurface to a relatively low pressure condition on the upper wing surfaceand that the prior art attempted to numerous techniques for dealing withinduced drag, but none are fully satisfactory.

While inventor accomplished some significant results in reducing draglosses of lifting foils, continual improvement in field is criticalgiven the increased commercial and governmental air transportation cost.Modern flow diagnostics applied to a very old aerodynamic problem hasproduced a number of intriguing new results and new insight intoprevious methodology and designs. Thus, any reductions in drag may on acraft provide savings.

SUMMARY OF THE INVENTION

This invention is different than that of the present inventor's priordisclosed design and yields an unexpected and improved stability andresult. Similar to the present inventor's prior design, there isprovided a lifting foil configured generally in the form of an endlessband having a plurality of exposed surfaces which meet at commonboundary lines. Here, however, the lifting foil comprises a generallyhorizontal and leading upper course, a generally horizontal and trailinglower course parallel to and spaced apart from the upper course and apair of opposed, vertically extending flow guides. A first one of theseflow guides joins a first end of the upper course to a correspondingfirst end of the lower course, while a second flow guide joins a secondend of the upper course to a corresponding second end of the lowercourse. The lower course configured with a main fuselage compartment.

The joinder of the upper and lower courses to the flow guides occurs atfour margins, referred to herein as an upper starboard margin, an upperport margin, a lower starboard margin and a lower port margin. The flowguides are blended into the courses at those margins to avoiddiscontinuities in the fluid flow across the inner and outer surfaces ofthe foil. In the use of the invention a working fluid, for example, butnot limited to water, air or other fluid particulate which may includesolids therein, flows from fore to aft through a large central passageand is first entrained by the leading upper course, where it exertsdynamic pressure outwardly against an exposed first surface of the foil.The working fluid also flows around the exterior of the foil, exertingdynamic pressure inwardly against a second surface thereof. The regionbetween the first and second surfaces defines a cambered lifting bodywhich reacts to the dynamic pressure on its first and second surfaces bygenerating a net upwardly directed lifting force. The lifting foil caninclude a third surface for dividing the large central passage into apair of smaller, spaced apart, passages having generally ellipticalcross-sections and a fuselage can be formed therewith.

With the leading upper course so positioned in connection with the flowguides angled to the rear to connect with the trailing lower course,there is diminished generation of initial spanwise fluid flow bybalancing the pressure at the upper starboard margin against thepressure at the lower starboard margin, and concomitantly balancing thepressure at the upper port margin against the pressure at the lower portmargin without any fluid interference caused by the trailing lowercourse housing the fuselage. The pressure balancing is accomplished byconfiguring the starboard flow guides to have a cross section which willreduce a dynamic pressure from a maximum value at the upper starboardmargin to a midpoint value of zero, while simultaneously increasing thedynamic pressure from a midpoint value of zero to a maximum value at thelower starboard margin. A similar process is carried out on the portside of the foil. Another aspect of the invention is to provide aleading upper course and a trailing course wherein the courses can havea range from positive to negative camber.

The lifting foil is unexpectedly provided with increased stability andcontrollability in flight. It is believed that the performance of thefoil is enhanced by virtue of a generally uniform upper leading courseproviding an initial fluid direction and further aided by flow guidesconfigured to stabilize the dynamic pressure at the ends of the upperand lower courses. This new design enhances the ability to achievereduced flow of fluid in the spanwise direction and reduces thegeneration of drag-producing vortices.

It is therefore an object of the present invention to provide anapparatus and method for improving the dynamic performance of a liftingfoil.

It is another object of the invention to reduce induced drag vortices tozero.

A further object of the invention is to reclaim vortex energy whilereducing tailplane aerodynamic balancing drag to zero.

A still further object of the invention is to enhance and increase liftwhile minimizing drag penalty.

Other objects of the invention are to enhance and increase lift anddecrease drag for very low speed flight, to reduce spanwise fluid flowin a lifting foil, to decrease aircraft drag and improve aircraftstability, as well as other and further objects and advantages as willbe apparent from the following description, the accompanying drawingsand the appended claims.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a perspective view of a lifting foil.

FIG. 2 is a top view of the lifting foil of FIG. 1.

FIG. 3 is a front view of the lifting foil of FIG. 1.

FIG. 4 is a side view of the lifting foil of FIG. 1.

FIG. 5A is a schematic illustration of pressure forces against a pair offlow guides.

FIG. 5B is a partially cut away perspective drawing of a flow guide.

FIG. 5C is a diagram of a cambered airfoil.

FIG. 6 is a schematic side view of a lifting foil according to theinvention.

FIGS. 6A-6I are cross sections of ribs located at angularly displacedpositions on FIG. 6.

FIG. 7 shows a parameterization of the model of FIG. 1 of the instantinvention is provided which includes a simplified box-wingconfiguration, defined using gap and stagger.

FIG. 8 is a perspective view of an alternative embodiment of theinvention.

FIG. 9 is a top view of the embodiment of FIG. 8.

FIG. 10 is a front view of the embodiment of FIG. 8.

FIG. 11 is a side view of the embodiment of FIG. 8.

FIG. 12 is a perspective view of yet another alternative embodiment ofthe invention.

FIG. 13 depicts cross sections A-C of ribs located at angularlydisplaced positions on FIG. 12.

FIG. 14 is a perspective view of yet another alternative embodiment ofthe invention.

FIG. 15 depicts cross sections B-D of ribs located at angularlydisplaced positions on FIG. 14.

FIG. 16 is a perspective view of yet another alternative embodiment ofthe invention.

FIG. 17 depicts cross section E of ribs located at angularly displacedpositions on FIG. 16.

FIG. 18 is a top view of FIG. 12.

FIG. 19 is a side view of FIG. 12.

FIG. 20 is a slightly turned side view of FIG. 12.

FIG. 21 is an end view of a propeller of FIG. 12.

FIG. 22 is an end view of FIG. 12.

FIG. 23 is a top view of another embodiment.

FIG. 24 is a top view of still another embodiment.

FIG. 25 is a top view of yet another embodiment.

FIG. 26(A-F) depict gap, stagger, decalage, dihedral, sweep andoverhang.

FIG. 27 depicts the effect of trailing tip vortex and curl.

FIG. 28 depicts the downwash, w, that decreases the geometric angle ofattack to the effective angle of attack, α_(eff), resulting in a dragcomponent called induced drag denoted by D_(i).

FIG. 29 depicts the effect of the downwash angle on lift.

FIG. 30 depicts a graph showing that increasing size winglets reducelift induced drag in comparison to flat wings.

FIG. 31 depicts a graph showing that when the gap increases (holdingspan constant) Munk's factor decreases and, consequently, CL decreases.

FIG. 32 depicts a graph showing span efficiencies for various optimallyloaded nonplanar systems.

FIG. 33 depicts a graph showing span efficiency.

FIG. 34 depicts a schematic of gap and stagger

FIG. 35(A-H) depict stagger and gap of an embodiment.

FIG. 36 is a screen capture of a window that was developed for the forcebalance application.

FIG. 37 depicts a graph of the level of hysteresis seen in the lift anddrag data with a model (gap 0.5 C, stagger (−) 0.5 C) at Re 60,000.

FIG. 38 depicts a graph showing a low level of Repeatability seen in thelift coefficient vs. angle of attack with a model (gap 0.5 C, stagger0.5 C) at Re 60,000.

FIG. 39 depicts a graph showing an example of the level of Repeatabilityseen in the lift coefficient vs. angle of attack with a model (gap 2.0C, stagger 1. C) at Re 120,000.

FIG. 40(A-B) depict Lift to Drag ratio Uncertainty Analysis at (a) at Re60,000 (b) at Re 120,000.

FIG. 41 depicts a graph of L/D vs. Decalage (at an angle of attack 5)and shows a small variation of the lift to drag ratio at varyingdecalage angles.

FIG. 42 depicts a graph of comparison of the Decalage effect between theAVL and NACA results show good agreement up until stall inception in theexperimental results.

FIG. 43 shows a graph of lift coefficients vs. Dihedral (at an angle ofattack 5) show comparatively little variation across a wide range ofvalues.

FIG. 44 shows a comparison of Dihedral effect between AVL and NACAresults shows good agreement.

FIG. 45 shows a graph of L/D vs. Sweep Angle (at an angle of attack 5)for different staggers shows adverse effects for the lift to drag ratioas sweep angle increases.

FIG. 46 shows CL vs. Overhang (at an angle of attack 5) shows adverseeffects for the lift coefficient as overhang ratio increases ordecreases.

FIG. 47 shows a comparison of Overhang effect between AVL and NACAresults shows good agreement.

FIG. 48 shows a comparison of AVL results shows no visible differencesbetween negative and positive stagger with Model #4 (1.0 C stagger, 0.5gap) and #6 (1.0 C stagger, 1.0 C gap).

FIG. 49 shows the variation of Lift coefficient with varying stagger (ata 5 degree angle of attack), the greatest variation is seen within 1chord length stagger with a sharp drop-off in variation beyond thatvalue of stagger.

FIG. 50 is a graph which shows the variation of Lift coefficient withvarying gap (at a 5 degree angle of attack), clearly the greatestvariation occurs within one chord length gap for most staggers.

FIG. 51 shows a comparison of gap and stagger effect between AVL andNACA results shows a similar result on biplanes

FIG. 52 shows a comparison of the AVL results of six parameters showsthat the gap and stagger have the most major effects out of the sixparameters.

FIG. 53 shows a CL comparison of AVL and UD Experiment with Model #1(1.0 C gap, no stagger) and Model #2 (0.5 C gap, no stagger) showssignificant variation between the experimental and AVL results; Model #2shows force balance measurements close to the AVL results.

FIG. 54 shows CL comparison of AVL and UD Experiment with the Model #3(0.5 C gap, (+) and (−) 0.5 C stagger) and Model #4 (0.5 C gap, (+) and(−) 1.0 C stagger) shows a large variation of lift coefficientcharacteristics between the negative and positive staggerconfigurations.

FIG. 55 shows CL comparison of AVL and UD Experiment with the Model #5(1.0 C gap, (+) and (−) 0.5 C stagger) and Model #6 (1.0 C gap, (+) and(−) 1.0 C stagger) shows a large variation of lift coefficientcharacteristics between the negative and positive staggerconfigurations.

FIG. 56 shows a CL comparison of AVL and UD Experiment with the Model #7(2.0 C gap, (+) and (−) 1.0 C stagger) and Model #8 (1.0 C gap, (+) and(−) 1.5 C stagger): the highest gap configuration, Model #7 does notshow any difference between the negative and positive configurations.The Model #8 shows a large variation of lift coefficient characteristicsbetween the negative and positive stagger configurations.

FIG. 57(A-C) show a comparison of lift and drag with varying stagger andconstant gap of 0.5 C at Re 60,000 shows that as stagger increases, theaerodynamic characteristics improve dramatically beyond an angle ofattack of 6°.

FIG. 58(A-D) show a comparison of lift and drag with varying stagger andconstant gap of 1.0 C at Re 120,000 shows that as stagger increases, theaerodynamic characteristics improve dramatically beyond an angle ofattack of 4°.

FIG. 59(A-C) show that as gap increases, the lift coefficient alsoincreases for a given angle of attack across all angles of attacktested.

FIG. 60(A-D) the highest L/D ratios for the highest gap for three casesshowing the best balance between additional parasite drag area andreduced lift induced drag due to the endplates.

FIG. 61(A-C) show the slope of the linear curve fit equations for CL asa function of varying stagger used to determine a generalized equationfor lift curve as a function of gap and stagger.

FIG. 62(A-B) show the collective lift curve slopes and Y-intercepts ofthe linear curve fits for the lift curve at three representative gaps todetermine a generalized equation.

FIG. 63 shows the position of the model (1.0 C gap and no stagger)relative to the laser plane of illumination.

FIG. 64 shows the downwash angle distribution in the spanwise directionshows that the biplane starts to produce a uniform downwash distributionat 12% of semi-span from the wing-tip.

FIG. 65 shows a schematic showing Downwash in a PIV velocity vectorfield.

FIG. 66 show streamlines of the biplane in 2D streamwise PIV usingTecPlot (at an angle of attack of 5°, (−) IC Gap).

FIG. 67(A-C) depict velocity distribution and downwash angle downstreamof Model #4 (0.5 C gap and (+) 1.0 C stagger) at different angles ofattack and a Reynolds number of 60,000.

FIG. 68(A-C) depict velocity distribution and downwash angle downstreamof Model #2 (0.5 C gap and no stagger) at different angles of attack anda Re 60,000.

FIG. 69(A-C) depict velocity distribution and downwash angle downstreamof Model #3 (0.5 C gap and (−) 0.5 C stagger) at different angles ofattack and a Reynolds number of 60,000.

FIG. 70(A-C) depict velocity distribution and downwash angle downstreamof Model #4 (0.5 C gap and (−) 1.0 C stagger) at different angles ofattack and a Reynolds number of 60,000.

FIG. 71 depicts Downwash angles at different AoA, 0.5 C constant gap, Re60,000 shows clear differences in upper and lower wing lift and clearfunctional dependency on stagger.

FIG. 72 depicts Downwash angle change and CL change at an angle ofattack of 10° as a function of stagger shows the change in downwashangle is proportional to the change in CL.

FIG. 73(A-C) show Lift coefficient estimation by downwash agrees wellwith the force balance measurement at the lower angles of attack and forthe (+) 1.0 C stagger model.

FIG. 74 shows an increase in Downwash gradient with increasing staggerand shows clear differences in slope before and after 5 degrees angle ofattack.

FIG. 75 shows flow visualization of 24″ Configuration of an embodimentof the instant invention at 50 mph (Re 215,000): Progression ofSeparation (reproduced from the reference with zero lift shifted angleof attack).

FIG. 76(A-C) show velocity distribution and downwash angle downstream ofModel #2 (0.5 C gap and no stagger) at different angles of attack and aRe of 120,000 shows increasing downwash angle with increasing angle ofattack.

FIG. 77(A-C) show velocity distribution and downwash angle downstream ofModel #1 (1.0 C gap and no stagger) at different angles of attack and aReynolds number of 120,000 shows increasing downwash angle withincreasing angle of attack.

FIG. 78 shows Downwash angles at different angles of attack, Re 120,000with no stagger shows that the upper wing has a higher downwash anglethan the lower wing.

FIG. 79(A-C) show the velocity distribution and downwash angledownstream of Model #4 (0.5 C gap and (−) 1.0 C stagger) at differentangles of attack and a Reynolds number of 120,000 shows increasingdownwash angle with increasing angle of attack.

FIG. 80(A-C) show the velocity distribution and downwash angledownstream of Model #6 (1.0 C gap and (−) 1.0 C stagger) at differentangles of attack and a Reynolds number of 120,000 shows increasingdownwash angle with increasing angle of attack.

FIG. 81 depicts a comparison of force measurement with Model #4 and #6shows a large CL variation at the angle of attack tested.

FIG. 82 depicts a Lift Coefficient at G=0.5 C, St=1 C, Re=60,000specifically shows a distinct change in lift curve slope.

FIG. 83(A-C) depict Downwash angle variation with increasing angle ofattack (0.5 C Gap, (+) 1.0 Stagger) and Re 60,000 shows a gradualincrement of downwash angle as angle of attack increases.

FIG. 84(A-B) depict a comparison of downwash angle between upper andlower wing shows a significant downwash change in the shaded region.

FIG. 85 depicts a side view of different stagger configurations showsdifferent planform area of the endplates acting upwash flow.

FIG. 86 depicts a DPIV image at 1 chord length downstream with a 9 m/s(Re 60,000) freestream velocity at 0 deg. angle of attack.

FIG. 87(A-C) show a comparison of the integrated force measurement tothe PIV derived momentum deficit drag for Model #2, 3 and 4. Blasiusflat plate drag and vortex drag are included as a reference.

FIG. 88(A-B) show a comparison of the integrated force measurement tothe PIV derived momentum deficit drag for Model #1(no stagger, 1.0 Cgap) and 6 ((−) 1.0 C stagger, 1.0 C gap) at Re 120,000. Blasius flatplate drag and vortex drag are included as a reference.

FIG. 89 shows a position of the models relative to the laser plane ofillumination.

FIG. 90(A-D) show a comparison of vertical velocity component on thevarying stagger configurations shows different velocity distribution atangles of attack of 0°, 5° and 8°.

FIG. 91(A-D) show a comparison of spanwise velocity component on thevarying stagger configurations shows different velocity distribution atangles of attack of 0°, 5° and 8°.

FIG. 92 is a pattern of the vertical velocity component of four varyingstagger configurations shows a large variation at an angle of attack 8°,and a Re of 60,000.

FIG. 93 shows the pattern of the horizontal velocity component of fourvarying stagger configurations shows a large variation at an angle ofattack 8° and a Re of 60,000.

FIG. 94 shows a view of the Trefftz plane with thee lines used tohighlight aspects of the flow structure around the wingtip and endplates

FIG. 95 shows a lift curve with varying stagger and a constant gap of0.5 C at a Re of 60,000 shows a significantly different lift coefficientcharacteristic of the (+) 1.0 C stagger configuration at an angle ofattack 8°.

FIG. 96 shows a comparison of the vertical velocity components with fourdifferent configurations at the upper horizontal line at an angle ofattack=8°, Re 60,000.

FIG. 97(A-C) show downwash angle distribution of the (+) 1.0 C staggerconfiguration shows a significantly different downwash characteristicthan other configurations at an angle of attack 10°.

FIG. 98 shows a comparison of the vertical velocity components with fourdifferent configurations at the lower horizontal line at an angle ofattack of 8° and a Re of 60,000 shows a different flow structure for the(+) 1.0 C stagger configuration than for other configurations.

FIG. 99 shows a comparison of the spanwise velocity components with fourdifferent configurations at the vertical line at an angle of attack of8° and a Re of 60,000 shows a different flow structure for the (+) 1.0 Cstagger configuration than for other configurations.

FIG. 100(A-D) show a vortex structure with varying stagger shows thatthe biplane with endplates generates different patterns of vorticity atα=0°, 5° and 8°.

FIG. 101 shows the Trefftz plane with two velocity slices used toinvestigate the vortex structure.

FIG. 102 shows a comparison of the vortex intensity at α=0°, 5° and 8°along a horizontal line across the vortex core with the no staggerconfiguration shows that the magnitude of the vorticity is directlyproportional to the angle of attack.

FIG. 103 shows a vortex core position selected relative to the wingtipand endplate.

FIG. 104 position of the models relate to the laser plane ofillumination.

FIG. 105(A-B) show a vertical velocity component with varying gap showsthat the biplane generates different patterns of vertical velocitydistribution at α=0°, 5° and 8°.

FIG. 106(A-B) show spanwise velocity component with varying gap showsthat the biplane generates different patterns of spanwise velocitydistribution at α=0°, 5° and 8°.

FIG. 107(A-C) show a vortex structure with varying gap shows that thebiplane with endplates generates different patterns of vorticity atα=0°, 5° and 8°.

FIG. 108 shows CL and downwash angle of the higher gap (1.0 C)configuration show higher CL than 0.5 C gap configuration.

FIG. 109(A-B) show a comparison of the vertical velocity components attwo different gap spacings at the upper and lower slice of velocities atan angle of attack of 8° and a Re of 60,000

FIG. 110(A-B) show a comparison of the magnitude of the vorticity andvertical velocity components with two different gap configurations atthe upper horizontal velocity and vorticity planar slices at an angle ofattack of 8° and a Re of 60,000 shows similar trends in the magnitude ofthe vorticity.

FIG. 111(A-B) show a vortex core position relative to the wing-tip andflow guide the vortex core position relative to the wingtip and endplatefor the two configurations at angles of attack of 5° and 8° and a Re of60,000

FIG. 112(A-D) show a comparison of the coefficients of lift determinedthrough integrated force measurement and circulation theory shows aclose agreement for the no stagger and positive stagger configurations,while somewhat difference for the negative stagger configurations.

FIG. 113 shows the position of the negative stagger models relative tothe laser plane of illumination

FIG. 114(A-B) show shows a comparison of the coefficients of lift by aforce balance and circulation theory

DESCRIPTION OF THE PREFERRED EMBODIMENTS NOMENCLATURE/ABBREVIATION

-   AP barometric pressure in mmHg-   AR aspect ratio-   a slope of lift curve-   acc acceleration-   a₀ two dimensional lift-curve slope-   b length of semi-span-   C_(L) _(U) lift coefficient on the upper surface of a biplane-   C mean chord length-   CD coefficient of drag-   CDo drag coefficient at zero lift-   CL lift coefficient-   C_(L) _(L) lift coefficient on the lower wing of a biplane-   C_(L) _(U) lift coefficient on the upper wing of a biplane-   Cf skin friction coefficient-   D_(end) _(—) _(plates) plates endplates drag-   D_(form) form drag-   Di induced drag-   D_(interface) interference drag-   D_(parasite) parasite drag-   D_(skinfriction) skin friction drag-   D_(total) total Drag force-   D_(wings) wing drag-   d_(P) diameter of the particle-   di image diameter of the particle-   e span efficiency factor-   f″ number of lens-   g gap ratio to the chord-   h perpendicular distance-   L lift force-   L/D lift to drag ratio-   l the length of the panel in the flow direction-   M magnification factor of the lens-   Ni image density-   RH relative humidity-   Re Reynolds Number-   S wing area-   SVP saturation vapor pressure for a given temperature T-   St stagger ratio to the chord-   U_(P) particle velocity-   u freestream velocity-   ū mean velocity-   u_(rms) root-mean-square velocity-   u(t) instantaneous velocity-   V freestream velocity-   v vertical component of the airflow-   w horizontal component of the airflow-   α angle of attack-   α_(i) lift induced angle of attack-   Γ circulation strength-   Γ_(z) circulation about z-axis-   ΔC_(L) _(U) additional lift coefficient on the lower surface of a    biplane-   Δt time interval-   ΔAz0 thickness of light sheet-   ΔH manometer reading-   ΔC_(L) _(U) additional lift coefficient on the upper surface of a    biplane-   ε downwash angle-   λ wavelength of light source-   μ dynamic viscosity of the fluid-   ρ density of the fluid-   ρ_(P) density of the tracer particles-   ρ_(red oil) density of the red colored oil in the manometer in kg/m3-   τ_(w) surface shear stress-   φ velocity potential (scalar)-   ω_(z) vorticity vector in two dimensional flow about z-axis-   ∇× curl-   ∇· divergence-   ∇ gradient-   U upper wing span b-   L lower wing span c chord-   CD total drag coefficient-   CDi induced drag coefficient-   CDiEM Munk's induced drag coefficient-   CDiOB Prandtl's induced drag coefficient-   CL lift coefficient-   LLC lower wing lift coefficient-   CLmax maximum lift coefficient-   ULC upper wing lift coefficient-   αLC lift curve slope e span efficiency factor-   EM Equivalent Monoplane-   g gap-   I electrical current-   k equivalent monoplane span factor-   L/D lift to drag ratio-   n load factor-   OB Orthogonal Biplane-   Oh Overhang Ratio (bU/bL)-   P power-   R A Distance Used in Calculating Downwash-   Re Reynolds Number-   s stagger-   S Reference Planform Area-   T thrust v induced velocity-   V induced velocity for a vortex segment-   V flow speed-   Vstall stall velocity-   W takeoff weight-   δ coefficient of mutual influence-   ρSL sea level air density-   σ angle of stagger-   σ stress

Referring now to the drawings, the present invention may have a form asgenerally illustrated by lifting foil 10 in FIGS. 1-4. FIGS. 8-11 alsoshow an embodiment of the invention and generally illustrated by thenumeral 10′. FIGS. 12-21 show alternative design propeller designsincorporating the novelties of the instant invention generallydesignated by the numeral 100. To these novel configurations it will beunderstood that the pitch of the courses can be similarly varied asdepicted in prior U.S. Pat. No. 7,100,867 incorporated herein byreference.

Using modern flow diagnostics applied to a very old aerodynamic problemhas produced a number of intriguing new results and new insight intoprevious results. The aerodynamic performance and associated flowphysics of the biplane with endplates as a function of variation in gapand stagger were analytically and experimentally investigated. Acombination of vortex lattice method, integrated force measurement,streamwise (Particle image Velocimetry) PIV, and Trefftz plane StereoPIV were used to better understand the flowfield around the biplane withendplates. A study was performed to determine the configuration with theoptimal aerodynamic performance and to understand the fluid mechanicsbehind optimal and suboptimal performance of the configuration.

FIG. 7 depicts a parameterization of the model of FIG. 1 of the instantinvention is provided which includes a simplified box-wingconfiguration, defined using gap and stagger.

The parametric study was performed using a Vortex Lattice code (AVL) forsix parameters of the biplane with endplates which equates to flowguides as used in the present invention gap, stagger, decalage,dihedral, sweep and overhang as seen in FIG. 26.

Gap is defined as the distance measured between the leading edges of thetwo wings perpendicular to the freestream (FIG. 26 a). Stagger isdefined as the distance measured between the leading edges of the upperand lower wings measured parallel to the freestream (FIG. 26 b) when theupper wing was offset in the forward direction, it has a positivestagger, according to the conventional definition. Positive and negativestagger configurations were considered.

Decalage on a biplane is the acute angle between the mean chord lines ofthe biplane as seen in FIG. 26 c. There are two different decalagedefinitions used: Aerodynamic decalage is the angle difference betweenthe zero lift lines of the two wings especially for cambered wings,Geometric decalage is the angle difference between the chord lines ofthe two wings without camber. In this study, since a symmetrical airfoil(NACA 0001) was used, geometric decalage is applied. Five differentcases with decalage, ranging from −4 to +4 with 2-degree increments, areconsidered in the AVL study.

Dihedral is defined as the upward angle from horizontal in a fixed-wingaircraft from root to tip, as shown in a front view in FIG. 26 d.Downward angled wings have negative dihedral, or anhedral.

Four different configurations were considered based on the shape of thewings:

-   Upper and lower wings in a dihedral position-   Both in an anhedral position-   Upper wing in a dihedral position and the lower wing in an anhedral    position-   Lower wing in a dihedral position and upper wing in an anhedral    position.

For each configuration, three different dihedral or anhedral angles areapplied with 2-degree intervals.

Wing sweep angles were considered for both wings of the simplified Houckconfiguration (FIG. 26 e). Three different cases, with sweep anglesranging from 0 degrees to 60 degrees with 30-degree intervals, wereevaluated.

Overhang ratio for a biplane is defined as the ratio of the span of thelower wing to the span of the upper wing. This is shown in FIG. 26 f.Overhang is written as:

${{Overhang}\mspace{14mu} {ratio}} = \frac{{lower}\mspace{14mu} {wing}\mspace{14mu} {span}}{{upper}\mspace{14mu} {wing}\mspace{14mu} {span}}$

Six different cases with overhang ranging from 0.8 to 1.2 wereevaluated.

AVL results show that the gap and stagger have the most dramatic effectsout of the six parameters studied for the biplane with endplate whenaspect ratio and the total wing area are held constant.

Other parameters considered for their influence on the aerodynamicperformance of the biplane configuration included dihedral, decalage,sweep and overhang. The effect of these parameters was observed to beeither negative or negligible. Decalage and dihedral under certainconditions could have a positive effect on the performance of thebiplane; however, these effects are comparatively small and wereneglected for the purposes of this study. Variation of overhang andsweep had a negative effect on the performance. An increase or decreasein overhang has a negative effect on the lift coefficient. From thevarious biplane configuration results obtained in AVL, the parametersfor wind tunnel testing were reduced to gap and stagger.

The difference in lift coefficients between positive and negativestagger configurations was just 0.01% through AVL analysis. Althoughthis is a non-physical result, it leads to the belief based on the AVLresults and Munk's theoretical results that negative and positivestaggered configurations with the same gap have the same aerodynamiccharacteristics. To better understand positive and negative staggereffects, different model configurations were investigated through windtunnel testing.

The specific results were obtained from force balance measurement.Fourteen biplane configurations with different gaps and staggers weretested in the wind tunnel at both Re 60,000 and 120,000. The forcebalance measurements show that as stagger increases in the positivedirection, the lift coefficient also increases. From the drag polar,there is a visible change in CDi across gaps with increasing gapproviding decreasing CDi. For the largest gap configuration, the highestL/D ratio is obtained near an angle of attack of 4°. As gap increases,the lift coefficient also increases for a given angle of attack acrossall angles of attack tested.

A large variation in lift behavior was found between the positive andnegative staggered configuration. A positive stagger can produce a CL47% higher than a negative stagger configuration for a 0.5 C gap, whichwas the largest variation. The lift coefficient has a weak dependence onReynolds number (the lift coefficient difference is less than 2.5%between Reynolds numbers 60,000 and 120,000).

A generalized empirical method for the prediction of lift coefficient asa function of gap, stagger and angle of attack has been determined andvalidated when combined with existing relations for CL-α adjustments forAR and taper effects. The resulting empirical approach allows for arapid determination of CL for a biplane having different gap, stagger,AR and taper without the need for a complete flowfield analysis.

Detailed insight into the fluid mechanical justification for the uniquecharacteristics of the biplane with endplates was determined in theexperiments through the use of the PIV method. Two Dimensional PIVresults show a distinctive pattern in the downwash angle for thedifferent gap and stagger configurations tested.

The downwash angle increases with increasing stagger. The positivestagger configuration has a higher downwash angle within the range ofangles of attack tested, while the negative stagger configuration has alower downwash angle. It is also evident that the change in downwashangle is directly proportional to the change in lift coefficient aswould be expected. Increasing gap spacing increases the downwash angleas well. A higher gap configuration has a higher downwash angle than thelower gap configuration. Like the stagger, the gap is proportionallyrelated to downwash angle. It is very important to note that thedownwash angles for the upper and lower wings were only the same at 0°angle of attack. The upper wing has a higher downwash angle than thelower wing for the all models tested at both Re 60,000 and 120,000. Thisdownwash angle variation between upper and lower wings increased withincreasing angle of attack. Based on the concept of downwash angle, itis appears that the upper wing in the biplane is responsible for agreater portion of the lift across a wide range of gap and stagger.

The downwash angle change was used to compute the additional liftcoefficient since the downwash angle is directly proportional to liftforce. This variation in downwash agrees well with the lift coefficientobtained through the force balance. These results differ greatly fromthose resulting from Munk's method for the calculation of additional CL.

Perhaps the most interesting behavior observed is that the lift slopefor an angle of attack range from 5° to 10° is greater than the slope ofthe lift coefficient for an angle of attack range from 0° to 5°. Thechange in the lift curve slope has been observed in the linear regime(−2°<α<8°) for all models tested in the University of Dayton Laboratory(UD LSWT). When increasing the stagger in the positive direction, thebiplane model experiences a positive change in the slope of the liftcoefficient. The lift slope for 5°<α<8° for the positive staggerconfiguration was significantly greater than for the negative staggerconfigurations. Downwash angle measurements using 2D PIV also show adramatic change in downwash angle in the angle of attack range between5° and 5.5° at a Re of 60,000.

The momentum deficit method was applied for the purpose of measuring thecomponents of drag force and parasite drag. The accuracy of this methodwas comparable to the force balance measurements; close agreement wasfound across the entire range of angles of attack tested, even thoughthe drag force by the momentum deficit method is a little lower.

According to Kutta-Joukowski theorem, the vorticity is proportionallyrelated to the lift force. The Stereo PIV analysis shows that thepositive stagger configuration has significantly different wakecharacteristics compared to the others. This implies that the endplatesof the biplane configurations control spanwise and vertical flow aroundthe wingtip differently. If the upper surface is positively staggered,the endplates can interfere with the flow field at the wingtip to reducethe spanwise flow over the upper wing. These spanwise induced velocitiesfrom the endplates oppose and thereby cancel those generated by theupper wing. Therefore, the spanwise and vertical velocity components canbe largely manipulated by the presence of the endplates and the staggercondition of the upper wing.

Stereo PIV analysis show how the wingtip vortex is formed for thedifferent stagger and gap configurations of the biplane with endplates.The stagger effect produces a large variation with respect to thevelocity components and the wingtip vortex structure. The negativestagger configurations show several vortices separated and spread out atthe lower wingtip and behind the upper wing. When the upper wing ispositively staggered, the biplane generates well-formed vortices. Thiscreates an upwash outboard of the endplates as well as a vortex at thetrailing edge of the wing and endplates.

From the observation of the vortex core location, the positive staggerconfiguration produces higher downwash and therefore, the lift forceobtained was higher than that obtained with the other configurations.The combination of force balance results for lift and downwash angleverify this flow behavior. For the gap effect, as gap increases, thiswingtip vortex roll-up behavior is similar but the magnitude of thevorticity for the larger gap configuration was higher. Therefore, as thegap increases for a given stagger condition a higher lift coefficient isobtained. When viewed at the same location the effect of gap did notchange the vortex morphology as much as the change in stagger.

Therefore, positive and negative stagger configurations produce asignificant difference in effective angle of attack for both the upperand the lower wings in the biplane. Different downwash angles wereobserved in terms of varying configurations for the upper and lower wingsurfaces in the biplane. The resulting measured integrated lift force isproportionally related to the observed downwash angles. This impliesthat one of the underlying assumptions in Munk's biplane analysis of theadditional lift force for upper and lower surface is invalid. A dramaticlift slope change (C_(L) _(α) ) was found at an angle of attack around5°. Clearly, positive stagger and larger gap configurations producehigher lift slopes than negative stagger and smaller gap configurationssince lift efficiency increased with increasing stagger and gap.

FIG. 1 is a perspective drawing of the lifting foil with a coordinatesystem attached. The coordinate system has its origin at thecenter-of-gravity of foil 10. It is right-handed and employsconventional 3-axis, X, Y, Z Cartesian coordinate designations.Directions may also be referred to as “starboard” (+Y), “port” (−Y),“fore” (+X), “aft” (−X), “up” (+Z) and “down” (−Z). The term “spanwise”is used to refer to sideward motion in either the +Y or −Y direction.The lifting foil 10 is preferably laterally symmetrical across the X-Zvertical plane. FIG. 2 is a top view of lifting foil 10 while FIGS. 3and 4 are front and side views respectively.

Lifting foil 10 comprises at least four basic elements, blendedend-to-end to form a closed loop surrounding an open interior. Theseelements are an upper leading course 14, a lower trailing course 15, astarboard flow guide 16 and a port flow guide 18. Upper leading course14 is positioned generally in an X-Y plane and has an upper starboardmargin 52 and an upper port margin 54. Lower course 15 is geometricallysimilar to upper course 14 and is situated parallel thereto. Lowercourse 15 has a lower starboard margin 56 and a lower port margin 58.Margins 52, 54, 56, and 58 are best shown by phantom lines in FIG. 1. Itis understood that foil 10′ is similarly formed in these respects.

Starboard flow guide 16 is joined to upper course 14 at upper starboardmargin 52 and is joined to lower course 15 at lower starboard margin 56.Port flow guide 18 is joined to upper course 14 at upper port margin 54and is joined to lower trailing course 15 at lower port margin 58. Itshould be understood that the margins 52, 54, 56 and 58 merely definethe geometric limits of the four principal elements and do not have anyparticular structural significance. Upper leading course 14 and lowertrailing course 15 generate lift in the +Z direction when traveling inthe +X direction through a surrounding fluid or when held stationaryagainst a surrounding fluid moving in the −X direction as depicted inFIG. 4. Flow guides 16, 18 are configured in the form of an arc having acenter portion 69 of substantially circular curvature and end portions68, 70 of a curvature which causes a blending with the surfaces of uppercourse 14 and lower course 15. (See FIG. 6).

Preferably upper course 14 is forwardly offset by a distance ΔX fromlower trailing course 15, as illustrated in FIGS. 2 and 4. As recentlydiscovered, this particular offset is a design feature which provides acargo carrying capability in or adjacent the lower trailing course 15appropriate for the intended use of the lifting foil 10. By way ofexample, FIG. 4 shows a side view of a cargo carrier 19 having asomewhat extended offset ΔX′. FIGS. 2 and 3 are a top view and a frontview, respectively of cargo carrier 19. The lower trailing course 15 canincorporate a fuselage 20 into the structure. As illustrated in FIG. 8,fuselage 20′ may be positioned midway between flow guides 16′ and 18′and extend upwardly from lower course 15′ to upper course 14′. Fuselage20′ may house a suitable power supply, controls and one or morepassenger compartments. Most conveniently a cockpit and crew quartersmay be located in a fore area designated by reference numeral 12 in FIG.2 (20′ in FIG. 8). Ample space is available for placement of aerodynamiccontrol surfaces. The entire surface of lifting foil 10 is preferablysmooth, and its four principal elements are blended together so as tominimize vortex generation. In a preferred embodiment lifting foil 10has a fuselage 20 within an interior region extending rearwardly andupwardly from reference point 12 (in FIG. 2) and exteriorly bounded bythe surfaces of the lower trailing course 15 in a central regionthereof.

In the embodiment of FIGS. 1-4, lifting foil 10 has a starboard flowpassage 22 and a port flow passage 24, as best illustrated in FIG. 3.These flow passages are generally symmetrical. As fluid flows throughpassages 22, 24, it experiences point-to-point variations in themagnitude and direction of the velocity. These velocity variations causevariations in dynamic pressure (commonly called “q”) throughout themoving fluid. According to the well known Bernoulli equation, q=½ρV2where: ρ is the density of the fluid, and V is the fluid velocity.

Lifting foil 10 experiences a net force having a magnitude and directiondepending upon the size of its wetted area and the variations in dynamicpressure thereacross. FIG. 5A illustrates a pattern of incrementalforces 63A-63H applied against the surfaces of flow guides 16, 18 as aresult of dynamic pressure variations. Incremental forces 63A-63H arerepresented by vectors oriented normal to the surfaces of flow guides16, 18 and having lengths corresponding to the magnitudes of theincremental forces being represented. For ease of illustration vectors63A-63H represent vector sums of incremental forces operating at thesame aspect angle against second surface 38 and third surface 39. Asused herein, reference numeral 63 applies generally to any of theillustrated vectors, specific ones of vectors 63 being designated 63 athrough 63 g. The total force exerted by the fluid against flow guide 16is given by the expression: F=ΣqΔA where ΔA is an elemental area wettedby the fluid. The vertical component of the vector F is the lift.

Still referring to FIG. 5A, attention is directed at vectors 63 a and 63b. These vectors are of equal magnitude, horizontal and oppositelydirected. Vectors 63 a and 63 b are therefore self-cancelling. Othervectors 63 have vertical components in addition to any horizontalcomponent. The horizontal components are self-cancelling, but thevertical components of these vectors are commonly directed. Thereforethe vertical components are additive, and lift-contributing. Similarlyport flow guide 18 is subjected to pressure forces represented byvectors 63 c and 63 d having self-cancelling horizontal components.Vectors 63 e 63 f represent pressure forces at margins 52, 56 of flowguide 16, for example and have no horizontal component.

As indicated previously, the inboard boundaries of starboard flow guide16 are delineated by upper starboard margin 52 and lower starboardmargin 56. These two margins are shown schematically in FIG. 5A by asingle, vertical phantom line. Similarly, a single vertical phantom linerepresents upper port margin 54 and lower port margin 58, the inboardboundaries of port flow guide 18. It will be understood that uppercourse 14 extends between upper starboard margin 52 and upper portmargin 54 and has a uniform pressure pattern shown by the length ofpressure vectors 63 f and 63 h. Similarly lower course 15 extendsbetween lower starboard margin 56 and lower port margin 58 and has auniform pressure indicated by vectors 63 e and 63 g.

Starboard and port flow guides 16, 18 create blended bridges betweenupper course 14 and lower course 15. And while their operation is notyet fully understood, it is believed that Starboard and port flow guides16, 18 connect courses 14, 15 together in such a way as to cause each tofunction as a terminator for the other. It appears that they provideequalizing fluid flows which balance out spanwise pressure variations,thereby making courses 14, 15 behave like virtual two-dimensional wings.However, regardless of the precise process taking place within starboardand port flow guides 16, 18, the lifting foil of this inventioninherently obtains remarkable results, as hereinafter described.

FIG. 5B illustrates a structure for a starboard flow guide 16. Thestructure of a port flow guide 18 would be identical, except for leftand right hand parts differences. Referring now to FIG. 5B, flow guide16 is seen to comprise a series of stiffening ribs secured betweensecond surface 38 and third surface 39. For additional stiffness, theillustrated ribs may be tied together by one or more spars (notillustrated). Surfaces 38 and 39 have been introduced above and serve ascovering for a large area of lifting foil 10. By way of example,surfaces 38 and 39 may be fabricated from thin aluminum sheet material.

FIG. 5B illustrates flow guide 16 with second surface 38 partially cutaway to reveal five typical ribs 83, 87. Two end ribs 81 and 89 are alsoexposed. These latter two ribs are welded or otherwise secured tostructure (not illustrated) at margins 52, 56. This creates blendedconnections between starboard flow guide 16 and adjoining ends ofcourses 14, 15. Two more ribs 82 and 88 are not illustrated in FIG. 5Bbut will be discussed below in connection with the description of FIGS.6B and 6H. Port flow guide 18 is similarly joined to courses 14, 15 atmargins 54, 58 respectively.

FIG. 5C serves as an aid in understanding the geometry involved in theachievement of coordinated lift at both upper and lower ends of flowguide 16. Shown there is a side view of a rib, being generic in natureand labeled for identification with reference numeral 64. Rib 64 has anupper surface 90, a lower surface 91, a chord line 92 and a mean camberline 93. The chord line runs between the ends of the rib 64, while themean camber line 93 follows a locus of positions midway between uppersurface 90 and lower surface 91. The area between chord line 92 and meancamber line 93 represents the lift available from a surface wrappedaround the rib 64. That lift may be increased by changing the curvatureof surfaces 90 and/or 91 to increase or decrease the area between chordline 92 and mean camber line 93. The lift may also be changed bymodifying the angle of attack, “a”, the angle between chord line 92 andthe relative fluid velocity “V”.

The operation of flow guide 16 is explained diagrammatically in FIG. 6.That figure has nine placement lines, 6A-6A through 6I-6I, indicatingthe locations and viewing directions for nine cuts across flow guide 16.Sketches of those nine cuts are appear on FIGS. 6A-6I, as indicated byreference numerals 81-89. Similarly, cross cuts across flow guide arerepresented by a, b, c in FIG. 13 and cross cuts across flow guide arerepresented by b, c, d in FIG. 15 and cross cuts across flow guide arerepresented by e in FIG. 17.

It is desired to produce substantially uniform lift along the lengths ofcourses, for example courses 14, 15 of FIG. 1 between their respectivemargin pairs 52, 54, 56 and 58. It is also desired to have that liftdrop substantially to zero at the course margins 63 a, 63 b, 63 c and 63d. Those desires substantially are met by placing flow guides 16, 18between courses 14, 15, at margins 52, 54, 56 and 58 as described above.The flow guides have cambered cross-sections which provide lift rangingfrom a maximum value down to a minimum value, preferably zero, and thenincreasing from the minimum value back to the maximum. The maximum valuefor the lift is the same as the value of the lift at margins 52, 54, 56and 58 of courses 14, 15. For present purposes it should be understoodthat the term “lift” is being used loosely as shorthand for “lift perunit span”.

FIG. 6 assumes that lifting foil 10 is moving in the +X direction,thereby creating an apparent fluid motion in the −X or aft direction andproducing lift in the −Z direction of FIG. 5B and +Z of FIG. 1, forexample. Also appearing on the figure is a vector diagram. Shown thereare the above-described net force vector, F, a lift vector, L and ahorizontal vector, H. V and L meet at an angle, “g”. H is perpendicularto L, so the three vectors define a right triangle. L is always verticaland is the vertical component of F. F may be positioned such that g mayhave any value between 0 and π radians. The camber of the cross-sectionsillustrated in FIGS. 6A 6E may be made proportional to COS(g), so thatthe lift generated by the first π/2 rad. goes from max to 0. However,the camber of the cross-sections illustrated in FIGS. 6E 6I should bemade proportional to COS(π−g), so that the lift over that angular rangegoes from 0 to max. This matches the terminating pressure conditions atmargins 52, 56. In a similar manner the pressure conditions at margins54, 58 are matched by flow guide 18. In this manner the presentinvention is able to avoid a negatively directed lift by either ofcourses 14, 15. This invention provides flow guides in matching pairssuch that each member views a termination of the other as a continuationof itself, thereby reducing pressure imbalances in the spanwisedirection.

Referring now to FIGS. 6A 61 there are shown, by way of example, ninerib cross-sections 81 89. Ribs 81, 89 are substantially identical andare cambered to provide the above described maximum lift. Rib 85 (FIG.6E) is perfectly symmetrical about a horizontal plane and generates nolift. Ribs 82 84 ramp the lift from the maximum down to zero, while ribs86 88 ramp the lift from zero back to the maximum.

Similarly, FIG. 12 shows multiple rib cross-sections a, b, c. Ribs “a”are substantially identical and are cambered to provide the abovedescribed maximum lift. Rib “c” is perfectly symmetrical about ahorizontal plane and generates no lift. Ribs “b” ramp the lift from themaximum down to zero and ramp the lift from zero back to the maximum.

Alternative embodiments of the foil design of the instant invention canbe seen in FIGS. 12-26. Similarly, there are provided, by way ofexample, rib cross-sections a-e in FIGS. 13, 15, 17 which correspond toFIGS. 12, 14, 16, respectively. FIGS. 14 and 16 provide comparativeembodiments to illustrate the invention.

It will be observed in FIG. 5C that the underside of lifting foil 10 iskeel-shaped, extending symmetrically upward at an angle “a” on bothsides of a baseline extending in the X-direction through reference point12. This provides a small dihedral for minimizing adverse effects ofsideslip. For embodiment as illustrated in FIG. 3, a suitable dihedralangle may be about 5 degrees, applied symmetrically to passages 22, 24.One of ordinary skill in the art may easily determine an appropriatedihedral angle for other embodiments of the invention. Cross sections oflifting foil 10 which are taken across the Y-Z plane have a generallyelliptical appearance. The elliptical shape seems to play a beneficialrole in the performance of the lifting foil. However, it is not knownhow much of the benefit is attributable to the elliptical shape and howmuch is due to other factors, such as flow guides 16, 18.

FIGS. 8-10 are top, front and side views respectively of a firstalternative embodiment of the invention, a cargo-carrying “stretch”version. This version is identical to the version of FIGS. 2-4, exceptfor the offset distance which has been increased from ΔX to ΔX′.

Referring now to FIG. 1, 8, 12, the empirical study discussed hereinindicates that the lifting foil of this invention is a substantialimprovement over the standard wing. In particular, there is asignificant improvement in span efficiency, in lift coefficient, inmoment coefficient and a stability margin.

Quite unexpectedly, the instant invention and configurations providedherein have provided a new insight to traditional understandings ofbiplane theory operation. The instant invention has been empiricallytested and is now the subject of a dissertation entitled “Gap andStagger Effects on the Aerodynamic Performance and the Wake behind aBiplane with Endplates” by Hantae Kang University Dayton, Ohio August2008. Part of the findings of Kang are incorporated herein in supportthe unexpected results and novel findings of increased aerodynamicsachieved by the instant invention.

A combination of vortex lattice method, integrated force measurement,streamwise Particle image velocimetry (PIV), and Trefftz plane StereoPIV were used to better understand the flowfield around the biplane withendplates. The Vortex Lattice code (AVL) shows that the gap and staggerhave the most dramatic effects out of the six parameters studied: gap,stagger, dihedral, decalage, sweep and overhang which runs in contrastto previously accepted aerodynamic performance and associated flowphysics.

Force balance measurements obtained with fourteen biplane configurationsof different gaps and staggers show that as gap and stagger increase,the lift efficiency also increases at all angles of attack tested atboth Re 60,000 and 120,000. Using the force balance data, a generalizedempirical method for the prediction of lift coefficient as a function ofgap, stagger and angle of attack was been determined and validated whencombined with existing relations for CL-α adjustments for AR and tapereffects. The resulting empirical approach allows for a rapiddetermination of CL for a biplane having different gap, stagger, AR andtaper without the need for a complete flowfield analysis.

Per the study, Two Dimensional PIV results show a distinctive pattern inthe downwash angle for the different gap and stagger configurationstested. The downwash angle increases with increasing gap and stagger. Itis also evident that the change in downwash angle is directlyproportional to the change in lift coefficient. Increasing gap spacingincreases the downwash angle as well. Based on the concept of downwashangle, the upper wing in the biplane is responsible for a greaterportion of the lift across a wide range of gap and stagger.

Stereo PIV analysis show that the positive stagger configuration hassignificantly different flow characteristics compared to the others. Thespanwise and vertical velocity components can be largely manipulated bythe presence of the endplates and the stagger condition of the upperwing. When the upper wing is positively staggered, the biplane generateswell-formed vortices. This creates an upwash outboard of the endplatesas well as a vortex at the trailing edge of the wing and endplates.

Therefore, positive and negative stagger configurations produce asignificant difference in effective angle of attack for both the upperand the lower wings in the biplane. Different downwash angles wereobserved in terms of varying configurations for the upper and lower wingsurfaces in the biplane. The resulting measured integrated lift force isproportionally related to the observed downwash angles.

A finding was made that the instant invention defies one of theunderlying assumptions in Munk's biplane analysis (Munk's 1923 paper“General Biplane Theory”) of the additional lift force for upper andlower surface is invalid. A dramatic lift slope change (C_(L) _(α) ) wasfound at an angle of attack around 5°. Clearly, positive stagger andlarger gap configurations produce higher lift slopes than negativestagger and smaller gap configurations since lift efficiency increasedwith increasing stagger and gap.

The integrated force study resulted in the conclusion that there is asubstantial difference between positive and negative stagger. Thisresult is a direct contradiction to the long standing predicted resultsof Munk. The integrated force study also exposed a “kink” in the liftcurve of the biplane with endplates. Apparently this “kink” has been inthe literature for over 75 years, yet no direct reference to it or itssource has been found.

Results from 2-dimensional and 3-dimensional flow diagnostic techniqueswere used to explain the differences uncovered in the force study inpositive and negative stagger, as well as the “kink” in the lift curve.The potential flow results do not show similar behavior; however, thereis very little dependence of the behavior of the “kink” on Reynoldsnumber in the range tested (60,000 to 120,000). Since the lift curve islinear with one slope before the kink, and again linear with a differentslope after the kink, suction lift is not suspected. In addition,2-dimensional streamwise momentum deficit drag accounts for the vastmajority of the 3-dimensional drag measured by the force balanceindicating the unlikelihood of significant rotational effects.

It was notable that the source of the “kink” in the lift curve is stillnot fully understood. It appears to always occur around the point ofmaximum lift to drag ratio. Therefore it is believed that whatever is atthe source of the kink increases lift faster than it increases drag.Many successful connections are made, nevertheless, between theintegrated force results and the flowfield results (usingKutta-Joukowski Integration of Lift, Momentum Deficit Drag, Downwashangle related to lift), and some related back to Munk's equivalentmonoplane and Prandtl's orthogonal biplane theorems.

As part of the empirical study, a model known now as the “HouckConfiguration” as generally depicted in FIGS. 1-4 was used for thestudy. The Houck Configuration is a type of biplane joined at the tipswith endplates or cambered flow guides 16, 18 designed to combine theindividual wingtip vortices into a single vortex that is more widespreadas displayed in FIGS. 5A and 5B. The study bears out the asserteddesign.

As noted in applicant's prior patent the described configurationasserted an improvement in the aerodynamic performance of the liftingfoil by reducing the induced drag through the reduction of spanwisefluid flow over the wings. If endplates reduce the lift induced dragmore than the associated increase in parasite drag then the L/D can beimproved as it is anticipated that the presence of endplates generatespositive interactions between the wings by capturing the tip vortices orreducing their deleterious effects.

An analytical and experimental examination of the aerodynamiccharacteristics of the biplane with endplates as a function of changesin several parameters commonly used to describe the biplaneconfiguration such as gap and stagger included:

Reducing the experimental parametric space of the biplane with endplateconfiguration through computational analysis using AVL (Athena VortexLattice Method, http://web.mit.edu/drela/Public/web/avl/); A largenumber of biplane related parameters varied using AVL and a set ofmodels will be chosen for experiments where large gradients in theaerodynamic performance were observed.

Measuring the integrated forces on the biplane configurations from windtunnel experiments to determine the aerodynamic effects of the relativeproximity of the second wing. Two dimensional velocity data in the wakeof the wing is generated using PIV; This is used to calculate the totaldrag on the biplane using the momentum deficit method to compare to theintegrated force. Three dimensional Stereo PIV was performed in theTrefftz plane in order to determine lift via Kutta-Joukowski integrationof circulation (a fundamental theorem of aerodynamics), which iscompared to the integrated lift force measurement.

Lift Induced Drag was also examined and induced drag is mainly createdby the vortices at the tip of an aircraft's wing. It is the “leastunderstood type of drag, but it is the most important, especially in thelow-speed region of flight”.

Induced drag increases proportionally as angle of attack increases. Thehigh pressure on the lower wing causes the airflow at the tips of thewing to curl around from bottom to top in a circular motion. Thisresults in a trailing vortex behind the wing tips. The circular motionat the wing tip creates a change in the angle of attack causing anincrease in drag. The more the angle of attack is increased, the greaterthe amount of lift developed and the greater the induced drag.

The lift slope for a finite wing decreases as the AR decreases. As theAR decreases, the flow effects over the wing due to the tip vortices arestronger. It was mentioned that wings with AR less than four have pooraerodynamic characteristics because they must be modeled by a largenumber of spanwise vortices. When the AR is increased it reduces theeffect of wing tip vortices. Therefore, wing tip vortices are animportant factor that must be considered to improve the aerodynamicperformance of any finite wing configuration.

One explanation for the physical mechanism for generating lift is theexistence of high pressures on the bottom surface and a low pressure onthe top surface of the wing. The net imbalance of the pressuredistribution creates the lift force and the induced drag results as aby-product. The airflow near the wing tips curls around the tips as itmoves from high to the low pressure regions. This circular motion iscalled wing tip vortices.

These wingtip vortices then cause a downward airflow in the wake of theaircraft at the trailing edge of the wing. This downward flow displayedin FIG. 27, acts the strongest toward the wing tip while losing strengthtowards the aircraft body assuming an elliptical distribution of lift.

Therefore, if wing tip vortices are controlled, the aerodynamicefficiency of the wing configuration will be improved. Wingtip vorticiesinduce a small downward component of air velocity at the trailing edgecalled downwash, w. FIG. 28 shows an inclined local airflow below thefreestream direction, V_(∞) by the induced angle of attack, α_(i).Downwash has two important effects on the wing performance: a reducedeffective angle of attack that reduces the lift, a portion of the liftthat acts as a drag force. A portion of the wing tip vortices cause arotation of the lifting force called induced drag, or drag due to lift.FIG. 28 also shows downwash, w, that decreases the geometric angle ofattack to the effective angle of attack, α_(eff), resulting in a dragcomponent called induced drag denoted by D_(i).

On the other hand, one way of trying to understand the amount ofdownwash produced by a wing surface is called the ‘momentum’ theory oflift. In this theory, the lift produced by a wing is equal to thedownward ‘push’ it gives to the air that it passes through. Bydeflecting the air downwards, the wing is lifted. This downwash angle isthe flow angle with respect to the relative wind (far field velocity).

From this relation it is observed that the downwash angle grows withincreasing lift coefficient. FIG. 29 shows a total lift force thatgradually increases with the downwash angle. Because this induced dragis a by-product of lift force, downwash angle at the tail is linearlyproportional to the lift force for the wing.

An elliptic wing with a 10% to 20% span extension winglet has a maximuminduced drag reduction of about 11% as compared with an ellipticallyloaded planar wing (see FIG. 30).

Munk in the General Biplane Theory derived some useful formulas usingtheoretical and experimental data. According to him, the additional liftcoefficient of staggered wing is:

${\Delta \; C_{L}} = {{\pm 2}C_{L}\frac{S}{b^{2}}\left( {\frac{1}{k^{2}} - 0.5} \right)\frac{b}{R}\frac{st}{b}}$

Where, S is the total area, st the stagger, b the span, k the equivalentmonoplane span factor, and R a distance used in calculating the induceddownwash. He gives

$\left( {\frac{1}{k^{2}} - 0.5} \right)\frac{b}{R}$

as a function of the ratio of gap to span G/b, which is called ‘Munkfactor’. (see FIG. 31).

It was noted that the span efficiency based on different nonplanar wingsshown in FIG. 32 denote the wake affects the minimum drag byillustrating the maximum span efficiencies for a range of concepts withfixed height and span. It is clear that the vertical extent of thesystem near the tips is the critical parameter.

A reduction of vortex drag may have a significant effect on fuelconsumption. Furthermore, vortex drag is even more significant at lowspeeds where vortex drag typically accounts for 80%-90% of theaircraft's climb drag at critical take-off conditions. The spanefficiency factor for the full endplate configuration (FIG. 33 was 22%improved over the nonendplate configuration). In addition, the 30% ofendplate also obtained around 10% drag reduction at CL=0.6 and more than22% span efficiency factor compared to nonendplate configuration.Interestingly, at CL>0.7, the 30% of endplate obtained a little higheraerodynamic characteristic than the full endplate configuration.

Kang describes aerodynamic forces acting on a biplane wing requires thecombination of several methods of analysis to explain thethree-dimensional characteristics of the flow with a focus on the vortexbehind a simplified Houck configuration. A simplified box-wingconfiguration, defined using gap and stagger, was used wherein Gap andStagger were varied. Gap is defined as the distance measured between theleading edges of the two wings perpendicular to the freestream. Staggeris defined as the distance measured between the leading edges of theupper and lower wings measured parallel to the freestream. When theupper wing was offset in the forward direction, it has a positivestagger, according to the conventional definition. The positive andnegative stagger configurations were considered.

Further, decalage, dihedral, anhedral, sweep were considered. Fivedifferent cases with decalage, ranging from −4 to +4 with 2-degreeincrements, are considered in the AVL study. Four differentconfigurations were considered based on the shape of the wings: Upperand lower wings in a dihedral position; Both in an anhedral position;Upper wing in a dihedral position and the lower wing in an anhedralposition; and

Lower wing in a dihedral position and upper wing in an anhedralposition. For each configuration, three different dihedral or anhedralangles are applied with 2-degree intervals.

Wing sweep angles were considered for both wings of the simplified Houckconfiguration. Three different cases, with sweep angles ranging from 0degrees to 60 degrees with 30-degree intervals, were evaluated.

Overhang ratio for a biplane is defined as the ratio of the span of thelower wing to the span of the upper wing. Overhang is written as:

${{Overhang}\mspace{14mu} {ratio}} = \frac{{lower}\mspace{14mu} {wing}\mspace{14mu} {span}}{{upper}\mspace{14mu} {wing}\mspace{14mu} {span}}$

Six different cases with overhang ranging from 0.8 to 1.2 wereevaluated.

Particle Image Velocimetry (PIV) was used to empirically measure aspectsof the Houck configuration.

-   -   a. Indirect velocity measurement; PIV measures the velocity of a        fluid element indirectly by means of the measurement of the        velocity of tracer particles within the flow.    -   b. Whole field technique; PIV method records images of large        parts of flow fields in a variety of applications in gaseous and        liquid media. Instantaneous image capture and high spatial        resolution allow the detection of special structures in unsteady        flow fields.    -   c. Velocity lag; this method requires us to check whether the        particles will faithfully follow the motion of the fluid        elements. Basically, smaller particles will follow the flow        better.    -   d. Illumination; a high power light source for illumination is        required such that the light scattered by the tiny tracer        particles will be recorded by the CCD camera sensor.    -   e. Duration of illumination pulse; the duration of the        illumination light pulse must be short enough to avoid blurring        of the image.    -   f. Time delay between illumination pulses; the time delay must        be long enough to be able to determine the displacement of the        tracer particles between two images and short enough to avoid        loosing particles with an out-of-plane velocity component,        leaving the light sheet between subsequent illuminations.    -   g. Distribution of tracer particles in the flow; A homogeneous        distribution of medium density is desired for high quality PIV        recordings in order to obtain optimal evaluation.    -   h. Density of tracer particles on the PIV recording; particle        density must be sufficient enough, so any loss of correlation        due to insufficient seeding is avoided.    -   i. Number of illuminations per recording; it should be        distinguished whether it is possible to store images of the        tracer particles on different frames for each illumination or        whether all particle images due to different illuminations are        stored on a single frame.    -   j. Size of interrogation area; this size must be small enough        that velocity gradients have no significant influence on the        results. It determines the number of independent velocity        vectors and the maximum spatial resolution of the velocity map.    -   k. Repeatability of evaluation; PIV recordings can easily be        exchanged for evaluation and post processing with others        employing different techniques.    -   A Neodym-Yttrium-Aluminum-Garnet (Nd:YAG) lasers laser        wavelength, α=532 nm) solid-state lasers for PIV were used in        which the beam is generated by Nd3+ ions for PIV. Talc was used        for seeding in this experiment, which is one type of magnesium.        A medium particle image density was used for standard        statistical PIV evaluation techniques.

The image density in an interrogation area is a result of the number ofparticles found in a circle diameter that has an area equal to aninterrogation area that is projected back into the airflow a distanceequal to the thickness of a laser sheet. It is thought that the requiredimage density that needs to be present to generate a high-validatedvector result in PIV expressed in particles per mm3, Ni.

Using the results of the AVL computational parametric study, twodefining factors for the aerodynamic characteristics of the biplane werechosen; gap and stagger. Eight different models were selected forexperimental wind tunnel testing based on these parameters. The modelswere fabricated according to the following gap and stagger variations.Each wing has a 12 inch semi-span and a chord of 4 inches with a flatplate profile and a thickness to chord ratio of 1.18%. Six models (Model#3˜#8) can be reoriented to produce positive or negative staggerallowing fourteen configurations to be obtained with eight models.

Parametric variation of stagger and gap by model number Model # GapStagger 1 1.0 C 0.0 C 2 0.5 C 0.0 C 3 0.5 C (+)0.5 C, (−)0.5 C 4 0.5 C(+)1.0 C, (−)1.0 C 5 1.0 C (+)0.5 C, (−)0.5 C 6 1.0 C (+)1.0 C, (−)1.0 C7 2.0 C (+)1.0 C, (−)1.0 C 8 1.0 C (+)1.5 C, (−)1.5 C * C: chord length(4 inches), AR = 6, total wing area = 192 in² for full span model

Semi-span models were used in a sidewall mount for greater force andmoment measurements to reduce error. Each model constructed was actuallya semi-span wing with bolts through the endplate for attachment. Thisallowed force measurements at this endplate.

The eight models have an average chord length (C) of 4″, a semi-wingspan(b) of 12″. The wings are based on a flat plate profile. They have arectangular planform with an area (S) of 192 in². The following are theSchematics of the full-scale models drawn in AVL (see FIG. 35A-H showeight models with different gaps and staggers).

Numerous experiments were performed in an Eiffel type low speed windtunnel at the University of Dayton (the UD LSWT). The wind tunnel is anopen circuit with the test section, 30 inches (width) by 30 inches(height) by 90 inches (length). The contraction ratio of the tunnel is16:1 and capable of continuous operation at velocities of over 120 mph.Aerodynamic forces and moments were measured at Reynolds number (Re)60,000 and 120,000 with a turbulence intensity of ˜0.035%. All modelswere mounted at the longitudinal mid-chord.

A six-component Gamma Sensor (ATI Industrial Automation) was used in asidewall mount configuration in the floor of the tunnel to measure theaerodynamic forces generated on each model. The maximum forces andmoments are listed in the following Table.

Table of Maximum Allowable Forces and Moments for the Balance ComponentMaximum Load Lift - FX ±235 lb Drag - FY ±235 lb Side - FZ ±736 lb Yaw -TX ±618 in-lb Roll - TY ±618 in-lb Pitch - TZ ±727 in-lb

The related sensing ranges and resolutions are listed in the followingTable. The calibration process for force and moment measurements wasperformed.

Table of Sensing ranges and Resolution (9-25 Calibration) Rated SensingComponent Ranges Resolution Lift - FX ±7.5 lb 1/2560 lb Drag - FY ±7.5lb 1/2560 lb Side - FZ  ±25 lb 1/1280 lb Yaw - TX  ±25 in-lb 1/1280in-lb Roll - TY  ±25 in-lb 1/1280 in-lb Pitch - TZ  ±25 in-lb 1/1280in-lb

On top of the force transducer, a rotary stage RM-5-100 from NewmarkSystems was placed for variation of the angle of attack. In order toobtain the data for a model, the test models are mounted on a rotarystage which is placed on the top of the force transducer. The forcetransducer and rotary stage were placed underneath the tunnel. Theaccuracy of the rotary stage is 72 arc seconds. All models wereconnected to the rotary stage by an aluminum cylinder through the floorof the tunnel.

The specifications of the rotary stage are listed in the followingTable.

Table of Rotary Stage Specifications Repeatability 5 arc-secondsResolution 0.36 arc-seconds Accuracy 72 arc-seconds Gear Ratio 72:1 Max.Load 275 lbs Moment 260 in-lb Max. Speed 1200 RPM Travel 360° Continuous

After the mounting mechanisms were installed in the wind tunnel, theaerodynamic forces and moments were measured before installing a modelat the desired wind tunnel speed. These are known as tare. The netaerodynamic forces and moments on the model wingspan sections aredetermined by subtracting the aerodynamic tare from the measured forcesand moments in the transducer.

Force Acquisition Software under the trademark LabView™ 8.0 was used fordata acquisition. FIG. 36 is a screen capture of a window that wasdeveloped for the force balance application in the UD LSWT.

The net forces and moments on the wing section were observed indirectlyby changes in voltage in strain gages. The strain gage values are theinput to the calibration matrix to obtain forces in lbf and moments inin-lbf. The sampling rate used was 1000 data samples per second. Datawere collected over two seconds in nonconsecutive one second timeintervals.

Determination of Freestream Velocity

Before any data were collected, atmospheric conditions at the tunnelinlet were obtained to normalize values collected at different times inthe study. The barometric pressure, humidity, and temperature in theroom were determined by barometer, hygrometer, and thermometer.

Two UAV relevant Reynolds numbers were selected, 60,000 and 120,000.Testing used angles of attack from −2° to 10°. The wind tunnel testsection velocity is set using a pitot-static tube and a manometer in thefollowing manner. The pitot tube is first placed in the freestream inthe inlet to the test section. The air density was calculated fromEquation for ΔH, where ρ is the air density, T is the room temperaturein Celsius, AP is the barometric pressure in mmHg, SVP is the saturationvapor pressure for a given temperature T, and RH is the relativehumidity.

$\rho = {1.2929\frac{273.13}{T + 273.13}\frac{\left\lbrack {{AP} - \left( {{S\; V\; P} \star {R\; H}} \right)} \right\rbrack}{760}}$

Substituting temperature, pressure and humidity into equation ρ yieldsthe local air density. The equation used to determine the delta H ininches needed for a given velocity is shown in the following Equation. Vis velocity in m/s, ρ_(air) is the air density in kg/m3, ρ_(redoil) isthe density of the red colored oil in the manometer in kg/m3, and g isgravity in m/s2, and the constant 39.37 is the conversion factor frommeters to inches.

${\Delta \; H} = {39.37\frac{V^{2}\rho_{air}}{2\; \rho_{redoil}g}}$

The Reynolds number equation was used for computing the freestreamvelocity with Reynolds numbers of 60,000 and 120,000. The velocitiesobtained from the Reynolds number equation were then used in equation3.2 to determine the manometer reading ΔH. This is how wind tunnelvelocity was set in all tunnel experiments.

Hysteresis and Repeatability

To verify the quality of the measured data, the models were tested forboth repeatability and directionality (hysteresis). An example of thiscan be seen in FIG. 37. At a 5° angle of attack, a mild hysteresis canbe seen in the lift data. This amount of hysteresis was indicativethroughout the tests at 5° angle of attack. 96.7% repeatability for Re60,000 and 95.3% repeatability for Re 120,000 were obtained in themeasured data FIGS. 38 and 39.

An aerodynamic tare is subtracted from the force balance measurement toobtain the net aerodynamic forces and moments. This tare is obtained bymeasuring the loads on the test mount without the models installed.

The turbulence level of the freestream was measured spatially using PIV.The intensity of turbulence is computed with the following equations;

${{{Turbulence}\mspace{14mu} {Intensity}} = \frac{u_{rms}}{\overset{\_}{u}}},{u_{rms} = \sqrt{\frac{1}{N}{\sum\limits_{i = 1}^{N}\left( {{u_{i}(t)} - \overset{\_}{u}} \right)^{2}}}}$

Where u(t), is instantaneous velocity, u_(rms) is the root-mean-squareof the turbulent velocity fluctuation, and ū is the mean velocity. Theturbulence intensity for the test area was found to be ˜0.035%.

Uncertainty Analysis

The uncertainty associated with the wind tunnel test measurement comesfrom a variety of components. This uncertainty arises from inaccuraciesinherent in the measuring tools and repeatability of measurements. Acomponent of error generally comes from two main sources: precisionerror (Px) and bias error (Bx). Precision error is the repeatability ofa measurement with a single part at the same environmental conditions.The precision of a testing device is therefore determined by takingmultiple measurements and determining the repeatability in themeasurements.

Uncontrollable effects, such as ambient temperature, humidity, andpressure, affect measurements with most instruments. This is bias errorthat shifts the measured value away from the true value. When all datapoints are shifted in one direction from the true value, bias error ispresent. Bias error is repeatable and is corrected by using a constantoffset value. Bias error remains constant when measurements are takenunder similar environmental conditions. In general, the uncertainty of ameasurement is expressed as a sum of precision and bias error as shownin the following Equation.

U=(B _(x) ² +P _(x) ²)^(1/2)

The total uncertainty of wind tunnel force measurements were calculatedby estimating the range of error resulting from both bias and precisionerror. This range of error is graphically displayed in Lift to Dragratio graphs.

The bias error was obtained from three different tunnel tests with thesame model. Although environmental conditions were manually checkedevery minute using a barometer, thermometer and hygrometer based on theprocedure herein, most experimental measurements can have uncontrollableerror factors. True values were estimated for each angle of attack asthe mean value calculated from the measurements recorded during threetunnel tests performed at different times.

The bias error was found for two cases: Re 60,000 and Re 120,000. CLdifferences in Table below represent the average difference of liftcoefficients measured in the three tunnel tests with the true CL value(mean value of the three test runs) for each angle of attack. Bias error(Bx) is the percentage of the CL difference to the true value for eachangle of attack.

Table for Bias error table for Re 60,000 and Re 120,000 Re 60,000 Re120,000 AoA CL CL (Deg.) difference bias error difference bias error 10.0070 12.3% 0.0098 11.4% 2 0.0086 6.5% 0.0059 3.4% 3 0.0101 4.9% 0.00361.4% 4 0.0116 4.2% 0.0048 1.4% 5 0.0131 3.7% 0.0079 1.9% 6 0.0146 3.4%0.0111 2.2% 7 0.0161 3.2% 0.0142 2.4% 8 0.0176 3.1% 0.0176 2.6% 9 0.01913.0% 0.0215 2.8% 10 0.0206 2.9% 0.0254 3.0%

The Gamma ATI F/T force balance was used to determine the forces on eachmodel. Precision error (Px) was computed using the standard deviation ofvoltage output from the force transducer. In general, standard deviationtells how much the data varies from the mean. 2000 samples werecollected during a nonsequential two second time period using acalibrated weight on the transducer with no wind tunnel airstreamflowing. The true value was considered as the mean value of 2000samples, which for this example is 4.358 and the standard deviation is0.003654 for the strain gage #1. In the same manner, Voltage differencesfor the other 5 strain gages were computed as displayed in the followingTable and the average precision error for all cases was computed as0.73%.

Table for precision error for each strain gage of Gamma sensor St.* #1St. #2 St. #3 St. #4 St. #5 St. #6 average Volt. 4.35 −2.26 4.54 2.16−7.53 0.47 mean Volt. 0.0036 0.0060 0.0029 0.0058 0.0020 0.023 diff Pre-0.37% 0.61% 0.30% 0.59% 0.20% 2.31% 0.73% cision error *St.: strain gage

Substituting bias and precision error into Equation for uncertainty ofthe tunnel testing data yields FIG. 40A-B which show the lift-to-dragratio plotted for Reynolds numbers 60,000 and 120,000 with error bars.

2-D PIV and 3D stereoscopic PIV systems were used to quantify thevelocity field around the models. To generate the instantaneous velocityfields the PIV system used included a high-resolution PCO 1600 CCD(charged-coupled device) camera, Nd:YAG laser (NewWave, 120 mJ/pulse), adelay generator (Quantum Composer), a laptop computer and a set ofoptics to generate the laser sheet in the test section at the desiredposition.

The experimental setup for the stereoscopic imaging system used torecord the PIV data. An angular displacement configuration (Scheimpflugcondition) has been used for the stereoscopic cameras. The Delaygenerator in the system generates two pulses in tandem with a specificinterval (typically 85 μs at Re 60,000) between the two pulses at arepetition rate of 10 Hz. The PCO 1600 camera used in the experiment hasan array size of 1600×1200 pixels. To generate 3D PIV, Scheimpflugadapters were used.

A powder seeder was used as a particle generator in the experiment. Thepowder seeder disperses the particles of talcum powder with an averagesize of about 1˜5 μm. Post-processing was performed on the 2D and 3D PIVdata using DPIVB version 2.1 developed by Innovative ScientificSolutions, Inc. (ISSI). The following are details of the subsystems forthis PIV setup.

High resolution of the camera determines the useful resolution of thePIV data. PCO.1600 Camera from The Cooke Corporation was used. It has a1600×1200 pixel monochrome CCD array with 1 GB of onboard memory. Thesystem features thermo-electric cooling (down to −50° C. vs. ambient)and noise level (down to 10 e-rms). The available exposure times rangefrom 5 μs to 49 days with 500 ns optional (see Table below).

Table showing Technical data of PCO1600 CCD camera Unit Setpoint PCO1600 Resolution (hor × ver) pixel normal 1600 × 1200 extended mode 1648× 1214 Pixel Size (hor × ver) μm2 7.4 × 7.4 Peak quantum % At 500 nmtypical 55 efficiency Full well capacity of e- 40,000 CCD Readout noisee- At 10/40 MHz 12/21 rms Image frequency, fps At full frame 30 framerate Exposure time s 5 μs~49 days (500 ns~49 days opt.)

One Nikon AF Micro Nikkor 55 mm lens for 2D PIV and two Nikkor 85 mmlenses for Stereo PIV were used with the cameras. The aperture was setat 3.5 and the magnification factor was 6.6. A field of view (FOV) withthis lens was varied based on focal distance, for example, at a focaldistance 80 cm with 55 mm lens, the FOV was around 13 cm.

Quantum efficiency is a quantity defined for the CCD as the percentageof photons hitting the photoreactive surface that will produce anelectron-hole pair. It is an accurate measurement of the device'ssensitivity. This figure indicates that the CCD sensors of the PCO.1600camera is most sensitive at a wavelength of around 500 nm thatcorresponds to the same wavelength of light from the Spectra PhysicsNd:YAG laser used for this study.

A dual laser-head system named “Solo PIV 120(made by Newwave)”, designedto provide a green light source for Particle Image Velocimetry (PIV)applications, is used in this study seen in FIG. 3.16. Two laser headswith 1064 nm output wavelength are mounted on a single base plate. Thebeams generated by these laser heads are combined and then enter asecond harmonic generator to produce green (532 nm) laser pulses. Theoutput of the second harmonic generator impinges on dichroic mirrorswhich transmit the residual 1064 nm energy to metal absorbers andreflect the 532 nm green energy to the laser exit port. The laser ispulsed from an external trigger source (Model 9614+ Digital Delay-PulseGenerator) through a rear panel BNC connector. Each externally suppliedpulse flashes the laser once at a repetition rate up to the maximum forthe model. Q-switch was controlled by means of an external triggersource (from the CCD camera control box) allowing precise triggercontrol of the laser energy pulse.

A Digital Delay-Pulse Generator (Quantum Composers, Model 9614) was usedto synchronize the Nd:YAG laser and CCD cameras. This provides fourprecisely timed logic transitions or two independent pulse outputs. Thetime interval, Δt, between laser pulses was controlled with the delaygenerator. During the time interval, Δt, particles move in and out ofthe laser light plane. Therefore, it is necessary to adjust thethickness of the laser light plane appropriately; 1 mm laser thicknessfor 2D PIV and 6 mm for 3D PIV were used in this study. The timeinterval depends on both the maximum particle displacement in theinterrogation area and the freestream velocity.

A powder seeding system designed by ISSI was employed in the study. Thissystem consists of 4 main components: body, spray tube, control box andan air compressor. Air pressure is adjusted by the control box todisperse seeding material, which was set higher than 20 PSI. Talcumpowder is put in the powder container that is placed in the body. Thepowder seeder is placed in front of the wind tunnel screens. Thisplacement of the powder seeder produces less interference than placementin or after the convergent section of the tunnel.

In PIV, a time interval (Δt) between two images depends on thefreestream velocity and the size of field of view. It is important toset a time interval because this significantly affects the measurementaccuracy. The Table below shows an example of a setup condition used todetermine the time interval. The figures show a freestream velocity of16 m/s and a field of view selected at 20.7 cm in the streamwisedirection. The time interval between laser pulses is set to 53 μs. Fromthis time interval, a 20% particle movement is observed across oneinterrogation area to achieve a high detection rate.

Table for Time interval example table Items Set Unit Pixels per ainterrogation area 32 pixel the number of vector arrows 50 on the axisfreestream velocity 16 m/s ROI (region of interest) 1600 pixel FOV(field of view) 20.7 Cm physical length of an interrogation 0.3 Cm areadelta T 53 μsDelta T: 53 microsecond−Particles moving: 10.5 pixels at 16 m/s provided(20% of the interrogation area 32×32 pixels)

Time sequences are coordinated to trigger by an external delay generator(Quantum Composers, Model 9614). Four independent timing signals betweentime intervals, T0, T1, T2, T3 and T4, are required for the bestperformance and maximum flexibility. Descriptions of each timing signalare in the Table below.

Table for Time set on the Delay-Pulse Generator Output of DelayGenerator Description T0 Time zero T1 T0 + 0 μs Flashlamp delay of thefirst laser T2 T1 + 122 μs Q-Switch delay of the first laser T3 T0 + 53μs Flashlamp delay of the second laser T4 T3 + 122 μs Q-Switch delay ofthe second laser

The strength of the Nd:YAG laser is controlled via the Q-switchsettings. If Q-switch mode is selected, the Nd:YAG laser delivers veryhigh power. An example of a time sequence to generate a ΔT of 53 μsshows, after time zero signal, T0, the flashlamp of the first laserbeam, delays for the time interval, T1 (0 μs).

The Q-switch pulse follows the first laser beam after 122 μs. In thesame manner, the flashlamp of the second laser operates after 53 μs. TheQ-switch pulse of the second laser operates after 122 μs. Cameraexposure time is coordinated using this time sequence to capture twoimages with the time interval desired. The time sequence to get dualimages is checked by the combination of a photodiode (Thorlabs DET200)and an oscilloscope.

The Digital Particle Image Velocimetry (DPIVB) software developed byISSI was used for post-processing and analyzes the velocity fields. Postprocessing was performed with an image correlation by adaptivemulti-pass on 32×32 pixels for an interrogation area with an overlap of50%. Because square-shaped interrogation areas were applied, a total of1813 displacement vectors (49 columns×37 rows) are obtained from 1600pix.×1200 pix. CCD array. With this setting, the physical length of aninterrogation area is 3 mm2 for a field of view 20 cm×15 cm. TecPlot360was used for vector display in color and additional post processing,such as plotting velocity distribution, vorticities, streamlines andcontour plots.

With the PIV method, seed particles influence how faithfully theyrepresent the local flow velocity. It is stated that there are two typesof error introduced into the PIV displacement fields. The first type oferror (type 1) is a completely random error. The amount of deviation forboth the vector direction and the magnitude are determined randomly, andthe locations of the outliers are also randomized throughout the entirePIV vector field. This type of error typically results in outliers thatare mostly surrounded by good vectors. It is asserted this type of errorusually occurs in a practical PIV flow if the noise on a correlationplane is erroneously identified as the signal peak. Another thoughtsuggests that the number of spurious vectors properly acquired in PIVdata should be around 5% of the total number of vectors in the field.The second type of error (type 2) is designed to produce clusteredoutliers. This type of error happens in a practical PIV flow mostlybecause of imperfections in the PIV image or low seed density. Type 2error is usually observed at the edges of the PIV frame.

For this study, more than 70 image pairs were collected to confirm thecross correlation method in obtaining the velocity vectors. Thisminimizes the PIV error factors. Outlier error vectors were eliminatedby filtering functions in the DPIVB software, such as: ‘Nearestneighbor’, ‘Log Linear LookUp (LUT)’. The ‘Nearest neighbor’ functioncompares and corrects each velocity vector with its nearest neighborscomputing the relative deviations in both magnitude and direction. The‘Log LUT’ function eliminates ‘blinking’ problems of particles as theypass by the periphery of the light sheet. When there is a brighterparticle among others in interrogation area, this particle affects nearneighbor particles. This function removes this effect. The standarddeviation from the PIV data in the freestream condition was 3.3%.

The calibration process is important for stereoscopic PIV because theimaging geometry generally causes significant aberrations and errorvectors. The calibration procedure will output a space and calibrationcoefficient map across the image where the calibration coefficients arerelated to the spatial distortion of the image. This distortion becomessignificant when imaging inside a wide fluid layer. Complex stereoscopicequations are used to output the calibration coefficients embedded inthe PIV software.

For calibration of the 2 cameras, a calibration target developed byDantec was placed in the center of the light sheet or object plane asshown in FIG. 3.22. The calibration target consists of 12×9 dots; eachdot has a 5 mm diameter and the depth of the concave semicircle is 4 mmwhich is used to calibrate x-direction movement measurement.

This girdded target allowed alignment of the camera positions in the xplane (freesteam direction) with a grid origin so that both cameras wereimaging a common object plane. Displacing the 3D calibration target inthe x-direction, allows the x, y, and z coordinate vectors to becalibrated with respect to the target. The calibration is accomplishedwith the two PCO1600 camera images with the target at two positionsalong the x-axis. Each camera in the real set-up acquires its image(FIG. 3.23), and then the calibration images are used to generate thevelocity vector distribution in the x, y and z axes in order toconstruct three-dimensional vectors through the DPIVB 2.1.

From the camera perspective, there is an image distortion differencefrom the left side of the image as compared to the right side in eachimage. The uniform, symmetrical calibration grid appears slanted, with adecreased vertical distance between rows on the inner sides of bothimages. Image alignment and distortion are corrected using polynomialequation mapping functions created using a least-squares method and thecalibration grid points using computer software (DPIVB). Each of the twocameras collects two-dimensional images in the wing wake that allowcalculation of two-dimensional velocity vectors. The two-dimensionalvectors from each image for each particle are compared for distortion todetermine a three-dimensional flow field (scaled in m/s). Vector colorin C indicates the magnitude of w (the lateral or z velocity component).The boxes connected by broken lines represent the same position in thecalibration images A, B, and C. Two-dimensional velocity vectorscalculated from DPIVB images are seen in images (B) and the finalthree-dimensional velocity vectors are viewed in the Trefftz plane in(C). The velocity vectors calculated for the image area are identifiedby the magnitude of their U, V and W components where U is the velocityvector in the streamwise direction, V is the velocity vector verticalcomponent of the airflow, and W is the horizontal component of theairflow. These values are used to calculate aerodynamic forces, such ascirculation, lift force, and lift induced drag.

Results of numerous tests conducted on different variations of thebiplane with endplates are provided. These results will be divided intothree main sections. The first part will show the results of the vortexlattice code (AVL) for six parameters of the biplane with endplates,which were described above: gap, stagger, dihedral, decalage, sweep andoverhang. This section uses the vortex lattice method to explain whichfactors have the greatest effect on the aerodynamic characteristics ofthe biplane configuration.

The second part will present the force balance data obtained in the UDLSWT. Based on the findings from the parametric study from the firstpart, fourteen biplane configurations with different gaps and staggerswere tested in the wind tunnel at Re 60,000 and 120,000. The results ofthe aerodynamic testing of Houck or Houck similar configurationsperformed by the Air Force Institute of Technology (AFIT), theUniversity of Maryland, the US Air Force Academy and NACA are comparedwith the UD flat plate profile biplane with endplates results.

The third section will offer fluid mechanical justification for theunique characteristics of the biplane with endplates determined in theaforementioned experiments. This detailed insight will be providedthrough the use of the PIV method. Total drag force computed by themomentum deficit method and the downwash angle for the biplaneconfigurations measured using 2D PIV will be shown. The 3D PIV data willbe presented showing the wing-tip vortex flow structure morphology inthe Trefftz plane. The lift force from vortices in the wake will becalculated using Kutta-Joukouski circulation theory.

Generalized dimensions were used for the computational analysis of thebox-wing configurations. The absolute dimensions used forexperimentation are a wingspan of 24 inches for each wing with a totalwing area of 192 in2 for both wings combined. This gave an AR=6[monoplane] for each wing. The AVL test conditions also included thedensity of air at sea level.

Decalage on a biplane is defined as the acute angle between the meanchord lines of the two lifting surfaces. Five different decalage anglesvarying from −4 to +4 with a 2-degree interval are considered. FIG. 41shows the results from AVL with increasing angle of attack. Results showthat L/D, for higher decalage angles, is also higher at lower angles ofattack.

These results were corroborated using the wind tunnel test results froma NACA Technical Report by Knight and Noyes published Wash. D.C., DTIC(Defense Technical Information Center) where good agreement was found atlower angles of attack. At higher angles of attack, results from AVLdeviate from the wind tunnel tests FIG. 42.

Results from AVL show a constant lift curve slope, whereas the slope ofthe lift curve from wind tunnel tests decrease as the stall angle isapproached, which is expected when viscosity is present. According toMunk's theory the effect in CL is observed as a result of a shift in theCL alpha curve with increase in decalage.

Dihedral Effect

The effect of Dihedral on the CL as a function of angle of attack isplotted in FIG. 43. Three of the four configurations tested resulted ineither no advantage or a disadvantage in the generation of lift from thewing configuration. A positive effect and the highest lift coefficientwere obtained for the biplane configuration with dihedral on the upperwing and anhedral on the lower wing. The total change in liftcoefficient found in variation of dihedral/anhedral was less than 6% andis insignificant compared to the changes due to other parametersevaluated.

A comparison of the wind tunnel tests and the results from AVL is shownin FIG. 44. For a large portion of the curves, the results are in goodagreement. Disagreement is observed when the assumptions behind thevortex method fail to consider the viscous effects in the flow, near theonset of stall.

Sweep Angle Effect

FIG. 45 shows L/D vs. angle of attack curves with varying sweep angles.Adverse effects on the performance of the biplane wing configurationwith increasing sweep angle are observed. These adverse effects increasenonlinearly as sweep angle is increased.

Overhang Effect

The overhang ratio for a biplane is defined here as the ratio of thespan of the upper wing to the span of the lower wing. FIG. 46 shows CLas a function of overhang ratio from AVL with varying overhang ratios.Lift coefficient results show a bell curve where the maximum CL isobtained at an overhang ratio equal to 1.

An increase or decrease in overhang ratio causes a decrease in L/D asdisplayed in the figure. A change in overhang ratio causes the endplatesto close in on one of the main wing surfaces, which in turn causes theoverall effective gap to be reduced. This effective decrease in the gapcauses the lift of the wing configuration to be reduced. As mentioned,it has thus far previously theoretically been described that the boxshaped wing configuration as the best configuration where the overhangratio of the wing is 1. One prior result shows that the highestaerodynamic efficiency is obtained from the box shaped wing.

FIG. 47 shows a comparison of the data from AVL to the wind tunnel data.The wind tunnel results show good agreement when superimposed over theAVL results. The slight variation can be attributed to the presence ofviscosity in the NACA wind tunnel results.

The Effect of Stagger

FIG. 48 shows a comparison between the negative and positive staggerconfiguration of two models using AVL. Lift coefficients at positive andnegative stagger can be seen to have very similar values. The differencewas just 0.01%. Although this is a non-physical result, it leads to thebelief based on the AVL results and Munk's theoretical results thatnegative and positive stagger configurations with the same gap have thesame aerodynamic characteristics. Clearly, AVL has some limitations indiscerning the measured differences in positive and negative stagger. Tobetter understand positive and negative stagger effects, different modelconfigurations were investigated through wind tunnel testing.

For the parametric study across the variables, 90 cases are selectedbased on a gap and stagger increment of 0.25 chord lengths as displayedin Table 4.1. Based on the AVL results of the negative and positivestagger configuration, these two stagger configurations were consideredas the same configuration in this AVL analysis.

Table of Case numbers with stagger and gap with 0.25 chord lengthintervals Gap 0 C 0.25 C 0.5 C 0.75 C 1 C 1.25 C 1.5 C 1.75 C 2 CStagger 0 C 1 2 3 4 5 6 7 8 9 (Positive 0.02 C 10 11 12 13 14 15 16 1718 or 0.25 C 19 20 21 22 23 24 25 26 27 Negative) 0.5 C 28 29 30 31 3233 34 35 36 0.75 C 37 38 39 40 41 42 43 44 45 1 C 46 47 48 49 50 51 5253 54 1.25 C 55 56 57 58 59 60 61 62 63 1.5 C 64 65 66 67 68 69 70 71 721.75 C 73 74 75 76 77 78 79 80 81 2.0 C 82 83 84 85 86 87 88 89 90 C:chord length

To study the effect of stagger, values of stagger between 0 and 2 chordlengths at increments of 0.25 C are considered. Results in FIG. 49 areplotted at values of constant gap for an angle of attack at 5 degrees.This graph shows the variation in lift coefficient with varying staggerat a 5 degree angle of attack.

This angle of attack was chosen as representative of all below stallangles of attack. In addition, 5 degrees angle of attack was selectedsince it is near the maximum lift to drag ratio angle of attack wherelift induced and parasite drag should be equal. The x-axis in the figurerepresents varying stagger from 0 to 2 chord length. The y axis showslift coefficient. The bottom solid line with triangular symbols is thesmallest gap and it shows the largest deviation. The highest line is thelargest gap and it shows the smallest deviation across the range ofstaggers. Also, it is clear that large variations continue up to 1 C,and very small changes occur beyond 1 C.

Munk's theory said that the change in lift due to stagger isproportional to the square of the stagger, and he stated this relationis exact enough up to ⅓ C stagger. For greater values of stagger equalto multiples of the chord, the dependence of lift on stagger is quitedifferent. At these values of stagger, however, some would considerthese to be no longer biplane configurations but more like tandemconfigurations. The change in lift of the upper and the lower wing isdirectly proportional to stagger, as long as it is stagger is small.

The Effect of Gap

The results of the computational study of the effect of gap on the liftgenerated are plotted in FIG. 50. It is observed that the liftcoefficient, at 5 degrees angle of attack, is a stronger function of thegap between the two wings of a biplane. Increasing the gap between thetwo lifting surfaces of a biplane will result in an increase in thetotal lift coefficient. A greater rate of increase in lift coefficientas a function of increasing gap is observed until the gap reachesapproximately 1 chord length distance. Above one chord length gap, therate of change of lift coefficient decreases with increasing gap.Further increases in the gap result in minimal interaction between thewings and leads to the lifting surfaces acting individually. Thus, froman air vehicle design perspective, any gap greater than a 1 chord lengthdistance between the individual wings of the box-wing configurationwould likely not merit consideration unless there were other designconstraints that drove the gap to a larger value.

FIG. 51 shows a comparison of the data from AVL to the wind tunnel databy another model. A gap of 0.75 C and three varying stagger ((−) 0.25 C,no stagger, (+) 0.50 C) were considered. The results from the windtunnel show good agreement with the results from AVL for the (+) 0.50 Cand no stagger configurations. The negative stagger configuration,however, has a different result: the CL of this configuration is thelowest from the NACA report. As mentioned, AVL has a restriction ingenerating stagger. Therefore, the inviscid limitation is expected to beresponsible for the difference between the two stagger configurations.

Selecting Two Parameters: Gap and Stagger

FIG. 52 displays a comparison of the AVL results of six parameters inthe generalized chord. This AVL results show that the gap and staggerhave the most major effects out of the six parameters studied for thebiplane with endplate configuration when aspect ratio and the total wingarea are held constant.

This AVL results show that the gap and stagger have the most majoreffects out of the six parameters studied for the biplane with endplateconfiguration when aspect ratio and the total wing area are heldconstant. Other parameters considered for their influence on theaerodynamic performance of the biplane wing configuration includedihedral, decalage, sweep and overhang. The effect of these parameterswas observed to be either negative or negligible. Decalage and dihedralunder certain conditions could have a positive effect on the performanceof the box wing configuration; however, these effects are comparativelysmall and were neglected for the purposes of this study.

Variation of overhang and sweep had a negative effect on the performanceat the Reynolds numbers tested. An increase or decrease in overhangcaused the effective gap between the two wings to decrease. Thisresulted in a negative effect on the lift coefficient. From the variousbiplane configuration results obtained in AVL, the parameters for windtunnel testing were reduced to gap and stagger.

Wind Tunnel Test Results CL Comparison of AVL and UD LWST Force BalanceMeasurement

Force balance data were obtained through UD LSWT testing. FIGS. 53, 54,55 and 56 compare lift coefficient variation with angle of attackmeasured during tunnel testing with the results from the Vortex Latticecode (AVL) for eight models in fourteen configurations at Re 120,000.Six models (Model #3˜#8) can be reoriented to produce positive ornegative stagger allowing fourteen configurations to be obtained witheight models. The AVL results for negative and positive staggerconfigurations are essentially the same (99.9%) as mentioned in theprevious section.

FIG. 53 presents CL as a function of angle of attack for Model #1 (1.0 Cgap, no stagger) and Model #2 (0.5 C gap, no stagger). CL MΔX wasobtained at an angle of attack 13° for model #1 and 14° for Model #2.For Model #1, experimental data show significant variation from the AVLresults; measured CL from the force balance was higher than predicted byAVL across the range of angle of attack until stall. Model #2 showsforce balance measurements close to the AVL results with gradual stallcharacteristics after 10° angle of attack. From the figure, increasinggap shows an overall improvement in lift performance.

FIG. 54 displays wind tunnel results for lift coefficient, CL, vs. α forModel #3 with 0.5 C gap oriented with positive (+) and negative (−) 0.5C stagger and Model #4 with 0.5 C gap oriented with (+) and (−) 1.0 Cstagger. In this plot, a strong dependence on stagger is clearlyobserved in lift coefficient. For the positive stagger configuration,both models have higher lift coefficient values than the AVL results.For the negative stagger models, the CL values are much lower than theAVL results. The lift slopes for both models in the negative staggerconfiguration drop significantly earlier than the models in the positivestagger configuration. The difference in the lift coefficient betweenpositive and negative stagger was most pronounced in Model #4 (49% at anangle of attack 15°). Model #4 in a positive stagger configurationproduces the most gradual stall characteristic for the angle of attackrange studied.

FIG. 55 shows the wind tunnel results for CL vs. α for Model #5 with 1.0C gap oriented with positive (+) and negative (−) 0.5 C stagger andModel #6 with 1.0 C gap oriented with (+) and (−) 1.0 C stagger. In thisplot, a strong relationship between stagger and lift coefficient is onceagain observed. For the positive stagger configuration, both models havehigher lift coefficient values than the AVL results. For the negativestagger model configurations, the CL values are only slightly higherthan the AVL results until stall. The lift slopes for both models in thenegative stagger configuration drop significantly earlier than themodels in the positive stagger configuration. The difference in the liftcoefficient between positive and negative stagger was most pronounced inModel #5 (26% at an angle of attack 13°). Model #6 in positive staggerconfiguration produces the most gradual stall characteristic CL plot forthe angle of attack range studied.

In the same manner, FIG. 56 shows CL vs. α for the Model #7 with 2.0 Cgap oriented with positive (+) and negative (−) 1.0 C stagger and Model#8 with 1.0 C gap oriented with (+) and (−) 1.5 C stagger. Unlike theprevious configurations, there is no visible difference observed untilstall between the negative and positive stagger configurations. It meansthat changing stagger no longer visibly affects CL for a gap of 2.0 C.

For Model #8, the positive stagger configuration has higher liftcoefficient values than the AVL results. For the negative stagger modelconfigurations, the CL values are only slightly higher than the AVLresults. Stall is approached more quickly for Model #8 in the negativestagger configuration than the model in the positive staggerconfiguration. For a better explanation of the stagger and gap effects,detailed analysis will be described in the following section.

Stagger Effects

Two sets of models with different gap and stagger were specificallyselected for further wind tunnel investigation. One set of models had aconstant gap of 0.5 C with varying stagger of (−) 1.0 C, no stagger, and(+) 1.0 C and all were tested at a Re 60,000. The other set of modelshad a constant gap of 1.0 C with varying stagger of (−) 1.0 C, nostagger and (+) 1.0 C and all were tested at Re 120,000. The resultsfrom the integrated force measurements are plotted in FIG. 57(A-C). FIG.57( a) is a plot of CL as a function of angle of attack. This indicatesthat as stagger increases in the positive direction, the liftcoefficient also increases with diverging results at an angle ofattack >3°. The maximum lift coefficient, CL max was observed at anangle of attack of 10° with the model in a negative staggerconfiguration. When stagger was zero or positive, a maximum CL max wasnot achieved before an angle of attack of 15°. Very subtle differencesin CL were observed at an angle of attack <5°, while significantdifferences were found as the angle of attack increased above 5°. Apositive stagger configuration produces a CL 47% higher than thenegative stagger configuration at an angle of attack of 11°. This isseen in the CL max of the negative stagger configuration. FIG. 57( b) isa plot of CD as a function of CL. From this figure, the minimum dragcoefficient, CDo-0.022 for three biplane configurations is observed.When the gap is held constant, as for the three models observed in thisstudy, similar CD values are expected. From the drag polar, there is avisible change in CDi across staggers with decreasing stagger providingincreasing CDi.

FIG. 57( c) shows the lift to drag ratios corresponding to FIGS. 57(a)˜(b). As stagger increases, the aerodynamic characteristics improvedramatically beyond an angle of attack of 6°. In the plots the highestL/D ratio is obtained around an angle of attack of 5°.

The second set of models had a constant gap of 1.0 C with varyingstagger of (−) 1.0 C, no stagger and (+) 1.0 C tested at Re 120,000.FIG. 58( a) is a plot of CL as a function of angle of attack. Theseresults indicate that as stagger increases in the positive direction,the lift coefficient also increases with diverging results at an angleof attack >6°. The maximum lift coefficient, CL max was observed at anangle of attack of 11° with the model in a negative staggerconfiguration. When stagger was zero or positive CL max was not obtainedup to an angle of attack of 15°. Very subtle differences in CL wereobserved at an angle of attack <6°, while significant differences werefound as angle of attack increased above 6°. A positive staggerconfiguration produces 26% higher CL than the negative staggerconfiguration at an angle of attack of 12°. This is seen in the CL maxfor the negative stagger configuration. FIG. 58( b) is a plot of CD as afunction of CL. From this figure, the minimum drag coefficient,CDo≅0.0182 for the three biplane configurations is observed. Also fromthe drag polar, there is a visible change in CDi across staggers withincreasing stagger providing decreasing CDi. According to the Munk'sfirst theorem, the induced drag of a multiplane lifting system isunaltered if any of the lifting elements are moved in the direction ofmotion. However, the tunnel testing of the biplane with endplates showsthat increasing the stagger in the positive direction produces apositive change in the slope of the lift coefficient and reduced liftinduced drag. This is a noticed empirical result.

FIG. 58( c) shows the lift to drag ratios corresponding to FIGS. 58(a)˜(b). As stagger increases, aerodynamic characteristics improvedramatically beyond an angle of attack of 4°. In the plots the highestL/D ratio is obtained around an angle of attack of 4°. Based on theforce balance measurements at Reynolds number 60,000 and 120,000, it wasfound that Reynolds number effects in this range were small. Thedifference between two Reynolds numbers was less than 2% as seen in FIG.58( b).

Gap Effects

Three models with 1 C of stagger were selected in an effort to isolatethe effect of varying gap: 0.5 C, 1.0 C, 2.0 C. These models were testedat Re 60,000 and Re 120,000. The results from the integrated forcemeasurements are plotted in FIG. 59.

In FIG. 59( a) CL vs. angle of attack, the results indicate that as gapincreases, the lift coefficient also increases for a given angle ofattack across all angles of attack tested. FIG. 59( b) shows CD as afunction of CL. From this figure, the minimum drag coefficient,CDo≅0.023 for all three configurations can be seen. Although each modelhas different gap spacing, there is no visible difference in the minimumdrag coefficient. The drag coefficient reduces as gap increases at agiven lift coefficient; the 2 C and 1 C gap configurations obtained morethan 29% and 17% drag reduction respectively compared to 0.5 C gapconfiguration at a CL=0.5 which is the maximum lift coefficient for thenegative stagger tests. From the drag polar, there is a visible changein CDi across gaps with increasing gap providing decreasing CDi. FIG.59( c) shows the lift to drag ratios. In these plots, the highest L/Dratio is obtained near an angle of attack of 4°.

The same models were tested at Re 120,000. FIG. 60( a) CL vs. angle ofattack indicates that as gap increases, the lift coefficient alsoincreases across all angles of attack tested. FIG. 60( b) shows CD as afunction of CL. From this figure, the minimum drag coefficient, CDo≅0.02for all three configurations is seen. Although all models have differentgap spacing, there is no visible difference in the minimum dragcoefficient indicating little contribution to the increased endplatedrag as endplate area increases with increasing gap. The dragcoefficient reduces as gap increases for a given lift coefficient; the 2C and 1 C gap configurations obtained more than 51% and 30% dragreduction respectively compared to 0.5 C gap at CL max for negativestagger. FIG. 60( c) shows the lift to drag ratios. In these plots, thehighest L/D ratio is obtained near an angle of attack of 4°. From thetunnel testing, it is obvious that increasing the gap on the biplanewith endplates increases the lift force and reduces the lift-induceddrag, improving the lift-to-drag ratio. Based on the force balancemeasurements of Reynolds number 60,000 and 120,000, it was found thatReynolds number effects for the varying gap were small across the rangetested. The difference between two Reynolds numbers was less than 2.5%as seen in FIG. 60( d).

A Generalized Method for the Prediction of Lift Coefficient

A generalized method for the prediction of lift coefficient as afunction of gap, stagger, aspect ratio and angle of attack has beendetermined empirically and subsequently validated. The development ofthe generalized equation begins with a linear curve fit for CL based onvarying stagger. Then varying gap was considered in the same manner. Thepre-stall CL data measured by force balance were used to create thegeneralized equation, to describe the linear lift curves. As explainedearlier, lift coefficient has a weak dependence on Reynolds number (thedifference is less than 2.5% between Reynolds numbers of 60,000 and120,000 tested). Thus it was considered unnecessary to include theeffect of Reynolds number in the generalized equation. Gap and staggerwere found to be the most relevant factors in the calculation of thelift curve.

FIG. 61 shows curve fits for CL as a function of stagger from (−) 1.0 Cstagger to (+) 1.0 C stagger in 0.5 C increments at different gaps.FIGS. 61( a) and (b) display a linear trend between stagger and liftcoefficient. As stagger increases, CL also increases. However, there isno difference observed between the negative and positive staggerconfigurations at a gap of 2.0 C, so the polynomial curve fit has noslope (see FIG. 61( c)). Thus at a gap of 2.0 C changing stagger nolonger affects CL.

FIG. 62 shows two examples of curve fits for CL as a function of varyinggap, for gap spacing of 0.5 C, 1.0 C and 2.0 C. The slopes andY-intercepts of the lift curves in FIG. 62( a) are obtained from FIG.61. As gap increases the lift curve slope decreases. At a gap of 2.0 C,however, the slope is 0. These slopes and intercepts are used in theformulation of a generalized equation as shown in FIGS. 62( a) and62(b).

The resulting equations for the curve fits are ((−0.0072·St+0.0145)·g)·αfor the slope and ((0.018·St+0.0499))·α for the Y-intercept. The twoequations can be combined to create a generalized equation for the liftas a function of angle of attack for a biplane with endplates as afunction of gap and stagger:

C _(L) _(GEN) =((−0.0072·St+0.0145)·g+(0.018·St+0.0499))·α

Considering aspect ratio effect, this equation can be rewritten asfollows:

C _(L) _(GEN)=(((−0.0072·St+0.0145)·g+(0.018·St+0.0499))·α)·(−0.0045·AR+0.0698·AR+0.7542)

where, St is the stagger, g is the gap and α represents angle of attack.Taper has a positive effect for lift; however, this effect iscomparatively small (2% difference for the entire range of taper from0.2 to 1) and was neglected. The resulting empirical approach allows fora rapid determination of CL for a biplane having different gap andstagger without a more extensive analysis. As validation of the accuracyof this equation, the agreement between the CL obtained by the forcebalance and the CL from the generalized equation is within ±7%.

Two Dimensional PIV Results

Three sets of models were selected to investigate the effect of gap andstagger using the 2D PIV method as represented in the Table below sincethe configurations have different gap and stagger. For stagger, set #1had varying stagger of 0C, (−) 0.5 C and (−) 1.0 C all with a (−) 0.5 Cgap. For gap, set 2 and 3 considered a gap of 0.5 C and 1.0 C with 0 Cand (−) 1.0 C stagger respectively.

Table for test conditions for 2D streamwise PIV For Stagger effect ForGap effect Set No. #1 #2 #3 Gap 0.5 C 0.5 C 1.0 C 0.5 C 1.0 C Stagger(+)1.0 C 0.0 C (−) (−) 0.0 C (−)1.0 C 0.5 C 1.0C Re 60,000 120,000120,000

Downwash angles were measured with 2D streamwise PIV techniques, 0 to 2C downstream from the trailing edge. All of the downwash angles weremeasured in a plane located at 33% semi-span in from the wing-tip on asemi-span direction to avoid the interference from the wing-tip andcorner intersection as seen FIG. 63.

FIG. 64 shows the downwash angle distribution in the spanwise directionfrom the wingtip to 16% of the semi-span wing for the lower wing of the0.5 C gap configuration. This stereo PIV result from the Trefftz planeshows that this rectangular planform biplane starts to produce a nearuniform downwash distribution at 12% of semi-span from the wingtip. Thisis validated by the fact that the downwash angle from streamwise 2D PIVfor this configuration was 2.90 at the same angle of attack of 5°.Hence, the 33% semi-span location used for streamwise PIV should be at arelatively uniform lift distribution location. To increase the dataaccuracy, more than 70 image pairs of 2D PIV images were used incalculating any averaged values.

Downwash Angle Calculation

This calculation is based on the downwash angle equation,

$ɛ = {\frac{a\; {{\tan \left( {v/u} \right)} \cdot 180}}{\pi}.}$

TecPlot360™ software was used to calculate this angle. It is importantto understand that downwash generally varies along the span of the wingand the measurements found in this study do not represent the averagedownwash angle across the entire wing span of each model. However,because all models have the same dimensions except the gap and stagger,this downwash angle is a representative value to understand flow physicsdownstream of the wing (see FIG. 65).

FIG. 66 shows a schematic of Downwash in a PIV velocity vector field. Inthis field, streamlines are parallel everywhere to the instantaneousvector field with two assumptions: 1) The flow is in a steady state 2)Particles follow the airflow faithfully.

The Stagger Effect on Downwash Angle

The stagger effect on the downwash angle was evaluated. Stagger wasvaried, 0C, (−) 0.5 C and 1.0 C lengths with a constant 0.5 C gap at aRe 60,000. FIGS. 67, 68, 69 and 70 show the velocity distribution anddownwash angle in the wake of the models with different staggers and aconstant gap of 0.5 C at angles of attack of 0°, 5° and 10°. Thedownwash angles were computed from the velocity vectors and are shown inthe Table below.

Table depicting Downwash angles at different angles of attack, 0.5Cconstant gap, and a Re of 60,000 Downwash Angle At AoA 0 deg. At AoA 5deg At AoA 10 deg upper lower upper lower upper lower Model #4 0 deg. 0deg. 4.6 deg. 3.3 deg. 13.4 deg.  9.8 deg. ((+) 1.0 C stagger) Model #20 deg. 0 deg. 3.9 deg. 3.2 deg. 9.9 deg. 8.3 deg. (no stagger) Model #30 deg. 0 deg. 3.6 deg. 3.0 deg. 7.2 deg. 6.8 deg. ((−) 0.5 C stagger)Model #4 0 deg. 0 deg. 2.9 deg. 2.5 deg. 6.6 deg. 6.2 deg. ((−) 1.0 Cstagger)

As seen in FIG. 71, these results show a distinctive pattern of thedownwash angle for different angles of attack. Downwash angle increaseswith increasing stagger. Model #4 with positive stagger ((+) 1.0 Cstagger, 0.5 C gap) has the highest downwash angle within the range ofangles of attack tested and Model #4 with negative stagger ((−) 1.0 Cstagger, 0.5 C gap) has the lowest. It was found that stagger isrelated, proportionally, to downwash angle. It is also evident that thechange in downwash angle is directly proportional to the change in liftcoefficient.

From the analysis, downwash angles for the upper and lower wings wereonly the same at 0 degrees angle of attack. This difference occurs sincethe lower wing is immersed in the induced downwash of the upper wing;hence, the lower wing operates at a lower effective angle of attack.According to Munk's theoretically based statement, if two wings of abiplane are identical, parallel, and unstaggered, the downwash producedby each wing is the same. He also described that “the condition ofminimum drag for biplanes calls for equal induced downwash over bothwings. That is the case only if the lift which produces the downwash isequal at both wings.”

However, the results of the present invention indicate differentdownwash angles for the upper and lower wings, regardless of the staggercondition. These differences become more pronounced at angles of attackaround 10 degrees. For all configurations, the difference in downwashangle between the upper and lower wing are small at an angle of attackof 5°. At an angle of attack of 10° a significant difference is seen.

FIG. 72 shows the comparison between the change in downwash angle andthe change in lift coefficient for the same models at a Re of 60,000. Asseen in the figure, as expected the change in downwash angle isproportional to the change in lift coefficient.

Using Munk's definition, the additional lift coefficient of staggeredwings is

${\Delta \; C_{L}} = {{\pm 2}C_{L}\frac{S}{b^{2}}\left( {\frac{1}{k^{2}} - 0.5} \right)\frac{b}{R}\frac{st}{b}}$

where, S is the total area, st is the stagger, b is the span,

$\left( {\frac{1}{k^{2}} - 0.5} \right)\frac{b}{R}$

is known as Munk's factor. According to Diehl, for the simplest biplanein which the wings are of same chord and span, the lift efficiency ofthe upper wing (or lower wing) differs from that of the biplane by anamount depending directly on the biplane lift coefficient. That is,

C _(L) _(U) =C _(L) +ΔC _(L)

or

C _(L) _(L) =C _(L) ∓ΔC _(L)

hence,

${{C_{L_{U}} + C_{L_{L}}} = {2\; C_{L}}},{C_{L} = \frac{C_{L_{U}} + C_{L_{L}}}{2}}$

where, C_(L), C_(L) _(U) and C_(L) _(L) are the lift coefficients forthe biplane, the upper wing, and the lower wing, respectively. When theupper and lower wings are of equal area, the increments ΔC_(L) _(U) andΔC_(L) _(L) are equal and of opposite sign. The Table below presents theadditional lift coefficient for the four different staggerconfigurations based on Munk's definition. Munk factors for the fourconfigurations are all 0.68 for the purposes of this comparison since

$\frac{gap}{span}$

is constant. This Table shows that the lift coefficient change for theupper and lower surfaces is the same, but with opposite sign. Also, forthe no stagger configuration, there is no change of lift coefficient onthe upper and lower wings.

Table showing Additional lift coefficient based on Munk's definition. AtAoA 10° C_(L) St (stagger) Munk ΔC_(L) _(U) ΔC_(L) _(L) Model #4 0.71  4in. 0.68 +30% −30% ((+) 1.0 C stagger) (63.4°) Model #2 0.552  0 in.0.68  0%  0% (no stagger)   (0°) Model #3 0.52 −2 in 0.68 +15% −15% ((−)0.5 C stagger) Model #4 0.51 −4 in. 0.68 +30% −30% ((−) 1.0 C stagger)(−63.4°) 

However, the experimental downwash analysis shows significantlydifferent results when compared to Munk's definition for the additionallift coefficients. This section addresses stagger effects on lift withrespect to the downwash angle. Model #2 with no stagger is used as abaseline to compare all other configurations.

In Equation 1.8, the lift coefficient is directly proportional to thedownwash angle,

${\frac{ɛ}{\alpha} = \frac{2\; C_{L_{\alpha}}}{\pi \cdot {AR}}};$

where, AR is constant for the four different stagger configurations.Therefore, the increments ΔC_(L) _(U) and ΔC_(L) _(L) can be estimatedwith downwash angle increments based on the Equation above.

C_(L) _(U) ≈C_(L)±Δε_(U), C_(L) _(L) ≈C_(L)∓Δε_(L)

where, Δε_(U), Δε_(L) represent the downwash angle change on the upperwing and lower wing respectively. The Table below shows the incrementsΔC_(L) _(U) and ΔC_(L) _(L) of the above models based on the results ofthe downwash angle change in the zero stagger configuration. Based onMunk's definition, for the zero stagger model there would be nodifference between ΔC_(L) _(U) and ΔC_(L) _(L) . However, the downwashangle at the upper surface is higher than the lower surface for the zerostagger configuration.

The Table shows the increments or decrements of Δε_(U) and Δε_(L) of themodels when varying stagger based on the zero stagger configuration. Forthe (+) 1.0 C stagger configuration, the Δε_(U) and Δε_(L) were 35.4%,18.1% respectively. As stagger moves to the negative direction, theΔε_(U) and Δε_(L) were significantly decreased. This trend of variationis in good agreement with the lift coefficient obtained through theforce balance. As displayed on the Table 4.4, the lift coefficient ofthe (+) 1.0 C stagger configuration is 28% higher than the no staggerconfiguration. The negative stagger configurations show lower CL thanthe no stagger configuration. These results are clearly in directconflict with those generated using Munk's definition for the additionalCL, explained in the Table immediately above.

Table showing Increments of Δε_(U) and Δε_(L) with varying stagger and afixed gap of 0.5C ε ε ε Δε Δε Δε at α = 0° at α = 5° at α = 10° at α =0° at α = 5° at α = 10° Model #4 upper 0° 4.6° 13.4° 0%   +18% +35.4%((+) 1.0 C lower 0° 3.3° 9.8° 0%   +3% +18.1% stagger) Model #2 upper 0°3.9° 9.9° 0 0 0 (no stagger) lower 0° 3.2° 8.3° 0 0 0 Model #3 upper 0°3.6° 7.2° 0%  −7.7% −27.3% ((−) 0.5 C lower 0°   3° 6.8° 0%  −6.3%−18.1% stagger) Model #4 upper 0° 2.9° 6.6° 0% −25.6% −33.2% ((−) 1.0 Clower 0° 2.5° 6.2° 0% −21.9% −25.3% stagger)

FIG. 73 shows the estimated change in lift coefficient based on thechange in downwash on the upper wing and lower wing with varyingstagger. The lift curve obtained from the force balance is compared tothis estimate. These downwash angles were measured at ‘0.33×semi-span’in from the wing-tip as mentioned earlier. It was therefore assumed thatdownwash distributions are similar at the same point along the wing spanwhen models have the same aspect ratio and same wing shape. As seen inthe figure, this estimate agrees well with the (+) 1.0 C configurationat the angles of attack tested and for the other models at lower anglesof attack. Under this assumption, the downwash angle increases asstagger increases.

In addition, the downwash angle change also increases as the staggerincreases towards positive values. This shows that the lift coefficientestimated by downwash agrees well in general to the force balancemeasurement at the lower angles of attack and for the (+) 1.0 C staggermodel.

The upper wing has a larger downwash angle than the lower wing. Model #4with positive stagger ((+) 1.0 C stagger, 0.5 C gap) has the biggestdifference in the downwash angle between its upper and lower wings at anangle of attack of 10°: 36%. The differences for Model #2, #3 and #4(negative stagger) were 20%, 7.5%, 11% respectively at an angle ofattack of 10°.

FIG. 74 represents the downwash angle gradient,

$\left( \frac{ɛ}{\alpha} \right),$

is a function of stagger. It is clearly observed that as the staggerincreases, the downwash gradient increases. Based on the concept ofdownwash explained earlier, the upper wing in a biplane is responsiblefor a greater portion of the lift force than the lower wing. Inaddition, interesting behavior was found in the downwash gradient. Theslope of the downwash variation with angle of attack was higher for anangle of attack range from 5° to 10° than the slope of the downwashangle variation with angle of attack for an angle of attack range from0° to 5°. This behavior corresponds to a kink observed in the lift curveslope determined through integrated force and will be explained furtherin the following section.

Video footage was taken with tufts at a Re of 166,000 across a range ofangle of attack from −6° to 10°. Results were zero lift angle shifted inorder to compare with test data, which was at an angle of attack of−2.14 at a Re of 125,000.

FIG. 75 shows when and where flow separation occurs on the wings. Itstarts first on the lower wing at an angle of attack of 4°. At an angleof attack between 8° and 10°, flow separation is observed on the upperwing. The tufts, at angles of attack from 5° to 10° are in randomorientations on the lower wing surface thereby demonstrating flowseparation. This observation supports that the upper wing in the biplanewith endplates is more responsible for generating lift force than thelower wing.

The Effect of Gap on the Downwash Angle

Downwash Angle with Zero Stagger Configurations

The effect of gap on the downwash angle will be explored in thesesections. Two different gap spacings with zero stagger were considered,0.5 C and 1.0 C at a Re of 120,000. FIGS. 76 and 77 show the velocitydistribution and downwash angle at angles of attack of 0°, 5° and 10°.In the same manner as in the previous section, downwash angles werecomputed from the velocity vectors and are shown in the Table below. Asseen in the Figures, these results show a different relationship for thedownwash variation with angle of attack than that seen with variation instagger. Increasing gap spacing increases the downwash angle. Model #1(no stagger, 1.0 C gap) has a higher downwash angle than Model #2 (nostagger, 0.5 C gap) in the range of angles of attack tested. As withstagger, the change in gap is proportional to the change in downwashangle.

As seen in Table below and in FIG. 78, it is very important to note thatthe downwash angles for the upper and lower wing were only the same at 0degrees angle of attack. Otherwise, the upper wing consistently has ahigher downwash angle than the lower wing; Model #1 (no stagger, 1.0 Cgap) has a larger variation in the angle between the upper and lowerwings than Model #2. The upper wing of Model #1 produces a 43% higherdownwash angle than the lower wing. Model #2 (no stagger, 0.5 C gap) hasa 35% difference in downwash angle between the upper and lower wings.This downwash angle variation between upper and lower wings wasincreased with angle of attack.

Table showing Downwash angles at different angles of attack, at nostagger, Re 120,000 Downwash Angle At α = 0° At α = 5° At α = 10° Model#2 upper 0 deg. 3.4 deg. 10.3 deg. (no Stagger, 0.5 C lower 0 deg 2.8deg.  6.7 deg. gap) Model #1 upper 0 deg. 4.2 deg. 12.7 deg. (noStagger, 1.0 C lower 0 deg. 3.1 deg.  7.3 deg. gap)

According to Munk's theoretically based statement, for two parallel andequal wings without stagger, the downwash angles of both upper and lowerwing should be the same. However, for all configurations tested, theupper and lower wing downwash angles were not the same. This impliesthat one of the underlying assumptions in Munk's biplane analysis forthe additional lift force for upper and lower surface is invalid.

Downwash Angle with (−) 1.0 C Constant Stagger Configurations at Re120,000

Two different Gaps, 0.5 C and 1.0 C with constant (−) 1.0 C stagger wereconsidered at a Re of 120,000. FIGS. 79 and 80 show the velocitydistribution and downwash angle at angles of attack of 0°, 5° and 10°.Similar to previously discussed, downwash angles were computed from thevelocity vectors and are shown in Table below.

As seen in the Table below, the downwash angles for the upper and lowerwing for models #4 ((−) 1.0 C stagger, 0.5 C gap) and Model #6 were onlythe same at 0 degrees angle of attack. Otherwise, the upper wing has agreater downwash angle than the lower wing; Model #6 ((−) 1.0 C stagger,1.0 C gap) has a larger downwash angle than Model #4, which was 28%larger than Model #4. This difference is very close to the difference inmeasured CL (25%), as seen in FIG. 81.

Table showing Downwash angles at different angles of attack, at (−) 1.0C constant stagger, Re 120,000 Downwash Angle At α = 0° At α = 5° At α =10° Model #4 upper 0 deg. 3.4 deg. 6.4 deg. ((−) 1.0 C Stagger, 0.5 Clower 0 deg 2.9 deg. 6.4 deg. gap) Model #6 upper 0 deg. 3.9 deg. 8.3deg. ((−) 1.0 C Stagger, 1.0 C lower 0 deg. 3.6 deg. 6.6 deg. gap)

From the three sets of downwash angle results, it is obvious that thedownwash angle of upper wing is larger than the angle of lower wing atboth Re 60,000 and 120,000. The downwash angle variation with angle ofattack of the upper wing is larger than the variation of the lower wing.Thus for the biplane with endplates, the upper wing generates more lift.Based on the concept of downwash angle, it appears that the upper wingin the biplane is responsible for a greater portion of the lift across awide range of gap and stagger.

Change in Downwash Angle Slope

A distinct change in the lift curve slope has been observed in thelinear regime (−2°<α<8°) for all models tested in the UD LWST (see FIG.82). This slope change occurred in other studies on the configuration ofthe instant invention.

When increasing the stagger in the positive direction, the biplane modelexperiences a positive change in the slope of the lift coefficient. Thelift slope for 5°<α<8° for the positive stagger configuration wassignificantly greater than for the negative stagger configurations. Thedownwash angle was investigated using the PIV method to betterunderstand this behavior using a smaller increment in angle of attackpreviously performed with Model #4 (gap 0.5 C and stagger 1.0 C). Thismodel has the greatest change in lift curve slope and it is hoped thatany change in the wake morphology would be easier to identify as aresult. FIG. 83 shows the velocity distribution and downwash angle atangles of attack ranging from 2° to 7° at a Re of 60,000. Between anangle of attack of 5° and 5.5° a dramatic change in downwash angle isseen which correlates to the measured lift coefficient change. Duringwind tunnel testing on this model, the highest L/D was achieved at thissame angle of attack.

The Table below displays the downwash angles for the upper and lowerwings for Model #4 (0.5 C Gap, (+) 1.0 C) as a function of angle ofattack at a Re of 60,000.

Table showing Downwash angles at different angles of attack with (+) 1.0C stagger, 0.5 C gap at a Re of 60,000 shows a significant downwashchange in the angle of attack 5.5°. Angle of attack 2° 3° 3.5° 4° 4.5°5° 5.5° 6° 7° Downwash angle 2.7 3.3 4.2 4.3 4.7 5 6.3 6.6 8.3 at theupper wing Downwash angle 2.4 3.2 3.4 3.7 3.8 3.8 5.3 5.7 6.3 at thelower wing

As the angle of attack increases, the downwash gradient increases aswell, especially, at angles of attack >5° (see FIG. 84).

These results correlate the force balance measurement explained in theprevious section well. In the plots of force measurement, the highestL/D ratio is obtained around an angle of attack of 5°. In addition, thelift coefficient also increases at roughly the same angle of attack.These results indicate that the angle of attack of 5° is a point ofinflection for a visible change in lift curve slope on the biplane wingwith endplates. Especially, since this configuration has positivestagger, the endplates can effectively interfere with the flow field atthe wing-tip to reduce the spanwise flow over the upper wing. Thiseffect of interference causes a dramatic increase of downwash angle atthe angle of attack of roughly 5°. This is also strongly related to thehigher lift force obtained from the tunnel test.

From the force balance measurements and streamwise PIV analysis, the gapand stagger have a substantial effect on the lift. One potentialexplanation for the lift variation based on the stagger could be themanner in which the different endplates act on the upwash around thewingtips of the biplane.

The effect of decreasing the endplate planform area is strongly relatedto the biplane wing efficiency. The endplates maintain theireffectiveness quite well until approximately 70% of the originalplanform area is removed. This means that the efficiency of the overallconfiguration for the cases of 100% and 30% endplate planform area isthe same. Therefore, in order to minimize skin friction drag andmaximize wing efficiency, the planform area of the endplates should beresized to 30% of its original chord.

FIG. 85 shows the side view of different stagger configurations: thepositive and negative 1.0 C stagger and the no stagger configurations.From the figure the planform area of the endplates extending upward fromthe lower wing is 50% of the endplate area for the (+) and (−) 1.0 Cstagger and 100% for the no stagger configuration. When a biplane isplaced in a freestream with an angle of attack, high pressure air rollsup from the lower wing to the upper wing. If the area of the endplate isdecreased, the skin friction drag acting on the upwash traveling alongthe surface of the endplates will be decreased as well. Therefore, the(+) 1.0 C configuration can have higher wing efficiency than the nostagger configuration. For the (−) 1.0 C stagger configuration, however,the last 50% of endplates area is placed downstream of the lower wing.Hence, there will be no endplate effect increasing the upwash magnitudefor the (−) 1.0 C stagger configuration.

One more potential explanation for different lift based on the staggercould be the wing-wing interaction between two wings of the biplane withendplates. From the force balance measurements, it was found that as gapincreases, the CL difference between the negative and positive staggerconfiguration decreases; Model #7 (2.0 C gap) sees almost no CL changeas a function of angle of attack between the positive and negativeconfigurations. When both wings are staggered, the fore wing reduces thewing efficiency of the aft wing because the aft wing is immersed in theinduced downwash of the fore wing; hence, the aft wing operates at alower effective angle of attack. Based on the concept of downwashexplained earlier, the upper wing in a biplane is responsible for agreater portion of the lift force than the lower wing. For the positivestagger configuration, the upper wing generates lift with lessinteraction from the lower wing but the lower wing will be subjected tothis interaction/interference effect. The downwash angle from PIV provesthis behavior. For the negative stagger configuration, the upper wing,which is the aft wing here will be affected by the lower wing, so thelift force generated will be lower than the positive staggerconfiguration. The PIV results prove that the downwash angle for theupper wing of the negative stagger configuration was smaller than thewith the positive stagger configuration. Therefore, because of thewing-wing interference effect, the positive stagger configuration cangenerate higher lift force than the negative stagger configuration.

Drag Force from 2D PIV Using the Momentum Deficit Method

Computation of the drag force is an important characteristic that canvalidate the PIV data generated during testing as well as correlate theintegrated force results. To measure the total drag force on the modelsat Re 60,000 and 120,000 with angles of attack of 0, 5 and 10 degrees,2-dimensional velocity field data were obtained using the 2D PIV method.The [non lift induced] drag for the biplane with endplates consists ofthree drag components: wing parasite drag, endplate parasite drag andinterference drag from the corner of the wing and endplate. Asdiscussed, the Momentum Deficit Method was applied to calculate the dragon both wings and the endplate. The averaged velocity field in thestreamwise direction between a half-chord length and one chord lengthdownstream was used in computing the wing drag force as displayed inFIG. 86.

Approximately 7% of the wing drag estimates the amount of interferencedrag and this was the technique used here. The range of the velocityfield in the streamwise direction between a half-chord length and onechord length downstream was used in computing the total drag force usingthe aforementioned momentum deficit method. Matlab software was used tocalculate the three components of total drag force. The total drag forcecomputed with Matlab is comparable to the drag force found from tunneltesting thereby validating PIV results. The frictional resistancecoefficient of the models was obtained using a Blasius method forlaminar flow. This was included to consider the skin friction drag ofthe endplates due to the inability of the PIV method to account forthese factors.

Two sets of models were used to compute and compare drag coefficients asshown in the Table herein. These three models consist of varying staggerof 0 C, (−) 0.5 C and (−) 1.0 C with constant 0.5 C gap. No significantdifference in total drag force is observed when comparing these models.Model #2 (no stagger and 0.5 C gap) has a slightly higher CD than theother two models but the difference was subtle.

Table showing Drag and CD with 3 models at Re 60,000 Model #2 Model #3Model #4 (stagger (stagger (stagger 0 C, gap (−) 0.5 C, (−) 1.0 C, 0.5C) gap 0.5 C) gap 0.5 C) AoA (deg.) 0 5 10 0 5 10 0 5 10 Wing 0.0620.105 0.235 0.068 0.102 0.267 0.065 0.102 0.257 Drag (N) End- 0.0030.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 plate Drag (N) Inter-0.005 0.008 0.017 0.005 0.007 0.019 0.005 0.007 0.018 ference Drag (N)Total 0.070 0.116 0.255 0.076 0.113 0.289 0.073 0.112 0.279 Drag (N) CD0.024 0.040 0.099 0.023 0.042 0.098 0.025 0.040 0.106

FIG. 87 shows the PIV derived drag data with the force balance drag dataat three different angles of attack. Vortex drag computed from StereoPIV data is plotted at angles of attack of 5° and 10°. Thevortex-momentum deficit contribution is 3.1% of the total drag. Inaddition, the Blasius equation for flat plate skin friction drag isplotted at the zero degree angle of attack for reference. This value isincluded due to the inability of the PIV method to take intoconsideration the flat plate drag of the endplates. Very close agreementcan be found at the observed angles of attack when the endplate flatplate drag estimate is taken into consideration.

In the same manner, FIG. 88 shows the PIV derived drag data with theforce balance drag data at different angles of attack for the higherReynolds number case. Close agreement can be found at the observedangles of attack between two methods when the endplate flat plate dragis taken into consideration.

Table showing Drag and CD with 2 models at Re 120,000 Model #1 Model #6(stagger 0 C, (stagger (−) gap 1.0 C) 1.0 C, gap 1.0 C) AoA (Deg.) 0 510 0 5 10 Wing Drag (N) 0.229 0.538 1.235 0.270 0.551 0.987 EndplateDrag 0.006 0.006 0.006 0.006 0.006 0.006 (N) Interference 0.017 0.0380.087 0.019 0.039 0.070 Drag (N) Total Drag (N) 0.252 0.582 1.328 0.2950.596 1.062 CD 0.022 0.050 0.116 0.025 0.051 0.105

The momentum deficit method was applied for the purpose of measuring thecomponents of drag force, parasite drag. The agreement between the twomethods was well within the degree of uncertainty, even though the dragforce by the momentum deficit method is a little lower for an angle ofattack of 10° for the 1.0 C gap configuration. Observing the velocityfield image at 100 angle of attack in FIG. 77 (for Model #1(no stagger,1.0 C gap) and 80 (for Model #6 (−) 1.0 C stagger, 1.0 C gap), apotential source of error can be seen with part of the velocity fieldcut off at the bottom of the figures.

Wing-Tip Vortex Structure of the Biplane with Endplates Using 3D PIV

Stereo PIV was used to study the flow physics in the Trefftz planebehind the biplane with endplates. To investigate the effect of gap andstagger from flow characteristics, two sets of models were selected forPIV testing with respect to the different gap and stagger as representedin the following Table.

Table showing test sets for 3D PIV For Stagger effect For Gap effect SetNo. #1 #2 Gap 0.5 C 0.5 C  1.0 C Stagger (+)1.0 C 0.0 C (−) 0.5 C (−)1.0 C 0.0 C Re 60,000 60,000 AoA 0°, 5°, 8° 0°, 5°, 8°Investigation of the Flow Structure with Varying Stagger.

To observe the structure of the flow, four different staggerconfigurations of (+) 1.0 C, 0 C, (−) 0.5 C and (−) 1.0 C stagger allwith a 0.5 C gap were used. The tests were performed at a speed ofapproximately 10 m/s, resulting in a Reynolds number of around 60,000.FIG. 89 shows the position of the models with respect to the laser lightsheet. The light sheet was located 0.82 chord lengths downstream fromthe longitudinal location in the middle between the upper and lowertrailing edge. The X, Y and Z axes represent the streamwise, thespanwise and vertical directions respectively.

The time interval of the laser pulses corresponding to the localdisplacement of the particles was 77 μs. The size of the field of viewwas 15 cm×10 cm using a 1600×1200 pixel CCD array. Three angles ofattack (0°, 5° and 8°) were selected for the Stereo PIV test.

FIG. 90 displays the comparison of the color contours of the verticalcomponent of velocity (v) with an overlay of v and w component velocityvectors. The black solid line in the shape of a ‘C’ represents thewing-tip area of the upper and lower wing with endplates of the biplaneas seen from behind. The scale of the figure is from −1.5 m/s to 1.5 m/swith 15 scales of resolution. The scale for the velocity vectors is heldconstant for each configuration and angle of attack. This allows foraccurate back-to-back comparisons between configurations.

The vertical component of velocity for Model #2 with the (−) 1.0 Cstagger and gap 0.5 C at angles of attack of 0°, 5° and 8° can be foundin FIG. 90( a). In the same manner, the figures for the (−) 0.5 Cstagger, no stagger and (+) 1.0 C stagger configurations can be found inFIGS. 90( b), (c) and (d). This shows upwash around the wing-tip whenthe endplates are connected at the upper and lower wing-tip of abiplane. The vertical component of velocity has a large variation withangle of attack. At an angle of attack of 0°, there is almost novelocity change on the upper and lower region of the wing surfaces forall models. This result is expected because a flat plate wing shouldgenerate no lift at α=0°. As angle of attack increases in all of themodels, the magnitude of the vertical velocity found outboard of theendplates gradually increases. In addition, there is a large variationin the patterns of the vertical velocity component across the models. Asillustrated in the figure, the biplane at angles of attack of 5° and 8°produces lift having high pressure air from the bottom of the wingescaping around the endplates to the top of the wing. This creates anupwash outboard of the endplates as well as a vortex at the trailingedge of the wing and endplates.

FIG. 91 compares the contours of the horizontal component of velocity(w) for the same configuration. The scale of the figure is from −1.5 m/sto 1.5 m/s with 15 scales of resolution. The scale for the velocityvectors is held constant across all angles of attack as was done for thestagger results.

The horizontal component of velocity for Model #4 with a negative 1.0 Cstagger at angles of attack 0°, 5° and 8° can be seen in FIG. 91( a). Inthe same manner, the figures for the (−) 0.5 C stagger, no stagger and(+) 1.0 C stagger configurations are plotted in FIG. 91( b), (c) and(d). These show spanwise flow around the wing-tip of the biplane. Thisvelocity component experienced in a large variation with respect toangle of attack. As the angle of attack is increased, the magnitude ofthe horizontal velocity on the upper surface gradually increases for allof the models. Also, large variations among the models can be seen.

The pattern of the vertical velocity component of the positive staggermodel is substantially different from the others, while the other threeconfigurations are very similar to each other as seen in FIG. 92.Because the upwash is causing the local airflow to travel along thesurface of the endplates, it is suspected that the vertical velocityvariation was due to the different stagger of the endplates and theirassociated rake angle.

The pattern of the horizontal component of velocity (w) for the sameconfigurations also shows a large variation as seen FIG. 93. Thepositive stagger model has a substantial difference from the others,while the other three configurations are very similar to each other atan angle of attack 8° and a Re of 60,000. It means that the wing-winginteraction between the upper wing and lower wing affects differentlybased on the stagger, although all of configuration have the same gapspacing. This wing interference effect for the flow field causes adramatic increase in downwash angle. This is also strongly related tothe higher lift force obtained from the tunnel test. Detailed velocityinvestigation was performed on specific linear slices along the Trefftzplane using the local velocity data as follows.

As seen in FIG. 94 three planes were used to compare the velocitycomponents of each model along the spanwise direction from z/c=−0.25 toz/c=0.25, which were the upper horizontal line, lower horizontal lineand middle vertical line. These were selected to highlight someimportant flow characteristics in the region surrounding the wingtip ofthe biplane.

FIG. 95 represents the force balance measurements explained in theearlier section. The lift coefficient was obtained at an angle of attackof 8° from the same configurations discussed above. As depicted in thefigure, four configurations show considerably different lift curvecharacteristics. The (+) 1.0 C stagger configuration producesapproximately 30% higher CL than the other configurations. This forcebalance measurement shows a very similar trend to the vertical andhorizontal flow structure since the vertical and horizontal velocitycomponents of (+) 1.0 C stagger configuration at an angle of attack 8°has a considerable difference from the others, while the other threeconfigurations are very similar to each other. This means the upwashstructure is directly related to the generation of greater lift.

FIG. 96 shows the comparison of the vertical velocity components withfour different configurations at the upper horizontal line at an angleof attack of 8° and a Re of 60,000. There is a large variation betweenthe curves. The curve of the (+) 1.0 C stagger configuration shows asignificantly different pattern than the others. The highest velocityseen with the configuration was (+) 1.75 m/s found outboard of thewing-tip and the lowest value was −1.0 m/s found inboard of thewing-tip. The asymmetry of this curve would indicate that the vortexmorphology is affected by the presence of the endplates. It is obviousthat the positive stagger configuration has significantly differentcharacteristics compared to the others.

A similar pattern of aerodynamic behavior for these configurations hasbeen observed in the downwash angle discussions evaluated by 2D PIV. Asdepicted in FIG. 97 the downwash angle at an angle of attack of 10° anda Re of 60,000 shows a considerably higher downwash angle than that seenwith the other three configurations. This behavior is very clearly seenin FIG. 97( c) for angles of attack >5°. These force balance data anddownwash angle measurements convincingly denote the flow behavior.

FIG. 98 shows the comparison of the vertical velocity components withthe same four configurations at the lower horizontal line at an angle ofattack of 8°, and a Re of 60,000. The graph shows distinct patterns foreach of the configurations. While three configurations (no stagger, (−)0.5 C and (−) 1.0 C stagger model) share similar behavior to that seenin the upper horizontal line, the (+) 1.0 C stagger configuration showssignificantly different behavior. This means that the positive staggermodel has a different flow structure forming around the lower wing area.This behavior is opposite to the flow structure observed on the uppersurface.

FIG. 99 compares the spanwise velocity components with four differentconfigurations at the upper horizontal line at an angle of attack of 8°and a Re of 60,000. This figure also shows significant differences forthe (+) 1.0 C stagger configuration. The maximum spanwise velocity forthe configuration was 1.75 m/s and at 48 mm from the wingtip. The graphof the (−) 1.0 C stagger configuration shows a maximum value of 1.3 m/sat 13 mm from the wingtip. The no stagger and (−) 0.5 C staggerconfigurations see 1.4 m/s at 23 mm and 1.1 m/s at 27 mm for the peakvalues and the locations respectively. Hence, the (+) 1.0 Cconfiguration produces greater spanwise flow at a higher location thanother configurations. This implies that the endplates of the biplaneconfigurations control spanwise flow around the wingtip in differentways. As discussed, endplates can hinder the spanwise flow and thusextend the wingtip vortices by causing a physical constraint to the flowfield. Extension of the wingtip vortices can cause a reduction ininduced drag.

This spanwise flow can be related to the aerodynamic efficiency based onthe force balance results and downwash angle. As displayed earlier, fromthe force balance measurement, this positive 1.0 C stagger configurationgenerated a 30% higher lift coefficient than others at the same angle ofattack. If the spanwise flow is formed far from the wingtip, this meansthat the endplates can spread out or splay the wing-tip vortices.Spreading out the wing-tip vortices could potentially cause a reductionin downwash and induced drag. Therefore, if the upper surface is staggerin a positive direction, the endplates can effectively interfere withthe flow field at the wingtip to reduce the spanwise flow over the upperwing. This spanwise induced velocities from the endplates oppose and canthereby cancel those generated by the upper wing. This is stronglyrelated to the 30% higher lift force the positive configuration obtainedfrom the tunnel test. Therefore, spanwise flow can be largely controlledby the presence of the endplates and the stagger condition of the upperwing.

FIG. 100 shows the vorticity for the four configurations at angles ofattack of 0°, 5° and 8° and a Re of 60,000. The scale of the figure isfrom (−) 200 rad/s to (+) 200 rad/s with 15 scales of resolution. Asillustrated in the figure, the biplane at different angles of attackgenerates different patterns of vorticity. As expected at an angle ofattack 0°, there is no observable vorticity. As angle of attackincreases, the magnitude of the vorticity gradually increases for eachmodel as would also be expected. Additionally, a large variation in theshape of the vortex between the models for a given angle of attack wasfound. The vortices are more ordered in the case of the (+) 1.0 Cstagger configuration. As described, this configuration produces thelargest spanwise velocity when compared to other configurations. Thefirst three configurations (no stagger, (−) 0.5 C and (−) 1.0 C stagger)show several vortices separated and spread out at the lower wingtip andbehind the upper wing. The wingtip vortices of the (−) 1.0 C staggerconfiguration are the least stable out of all four models. This causesthe lowest averaged spanwise velocity and also the lowest liftcoefficient from the force balance measurement at an angle of attack 8°.The first two configurations, which are (−) 0.5 C and (−) 1.0 C staggerconfigurations show scattered wingtip vortices near the wingtips andendplate. As described earlier, broadening of the wingtip vortices cancause a reduction in downwash and induced drag. From the drag polarforce measurements, there is a visible change in CDi across stagger withincreasing stagger providing decreasing CDi.

Behavior of the vortex roll-up past the positive 1.0 C staggerconfiguration is different to others. Different pattern of vortexroll-up came from the different pattern of spanwise and verticalvelocity components of this configuration, which have visual differencescompared to other configurations. This different pattern of wing-tipvorticies causes significantly different downwash angle and liftcoefficient from the force balance measurements.

Planar velocity slices similar to the ones used in the previous sectioncan be seen in FIG. 101, two planes were used to compare the vorticitycomponents of each model along the spanwise direction from z/c=−0.25 toz/c=0.25: a horizontal line in the vortex core and a vertical line inthe vortex core. These were selected to show the most remarkable flowcharacteristics at the wingtip.

FIG. 102 shows the comparison of the magnitude of the vorticity withrespect to angle of attack at 0°, 5° and 8° at the horizontal line ofthe vortex core. This figure shows that the magnitude of the vorticityis directly proportional to the angle of attack.

FIG. 103 shows the vortex core position relative to the wingtip andendplate for the four configurations at angles of attack of 5° and 8°and a Re of 60,000. The coordinates of the vortex core are normalized bythe chord length. The circle symbol on the figure represents the vortexcore position at α=5° and the triangle symbol is for α=8°. The vortexcore is the region of low pressure. As angle of attack increases therewere small changes in the vortex core location with the exception of the(−) 1.0 C configuration, where the vortex magnitude was significantlydifferent. Actually, it is difficult to observe the vortex core for the(−) 1.0 C and (−) 0.5 stagger configurations because of the splay of themultiple vortices. For the first three configurations, the vortex corelocations were selected around the middle of the corner of the lowerwing. Unlike the other configurations, the (+) 1.0 C stagger model has afundamentally different vortex core location, farther up the upperwingtip. This roll-up behavior produces stronger vortices which causes achange in shape of the downwash distribution, which was displayed inFIG. 97. As explained, the lift force produced by the wings is equal tothe downward ‘push’ it gives to the air that it passes through.

Investigation of the Flow Structure with Different Gap Configurations

To observe the flow structure around configurations with different gapspacing, two different configurations of 1.0 C and 0.5 C gap with nostagger were used. The tests were performed at a speed of approximately10 m/s, resulting in a Reynolds number of approximately 60,000. FIG. 104shows the position of the models with respect to the laser plane ofillumination. The light sheet was located 0.82 chords downstream fromthe upper and lower wing trailing edge. The X, Y and Z axes representthe streamwise, the spanwise and vertical directions respectively.

Similarly, the time interval between the laser pulses corresponding tothe local displacement of the particles was 77 μs. The size of the fieldof view was 15 cm×10 cm and three angles of attack (0°, 5° and 8°) wereselected for the Stereo PIV test. FIG. 105 displays the comparison ofthe magnitude of the vertical and spanwise components of velocity (v)for Model #1 (1.0 C gap) and #2 (0.5 C gap) with no stagger. The biggerblack solid line in a ‘C’ shape represents the wing location of theupper and lower wing of 1.0 C gap. The scale of the figure is from −1.5m/s to 1.5 m/s with 15 scales of resolution. As seen in the Figure, thepattern of vertical velocity component is very similar between the twoconfigurations. The highest vertical velocity component was observednear the corner area of the lower wing for both configurations. Thisflow pattern shows upwash moving up along the endplates when the upperand lower wingtips of a biplane are connected with an endplate. Thisvertical component of velocity experiences a large variation with angleof attack. At an angle of attack of 0°, there is almost no velocitychange on the wing and endplate surfaces for all models as expected. Asangle of attack increases, the magnitude of the vertical velocity foundoutboard of the endplates gradually increases up to the tip of the uppersurface. There is little variation in the overall pattern of thevertical velocity components between the two models, but the scale ofthe flow contour was different based on the gap spacing. As illustratedin the figure, for the case of angles of attack of 5° and 8°, flow fromthe bottom of the wing moves around the endplates to the top of thewing. This creates an upwash outboard of the endplates as well as avortex at the trailing edge of the wing and at the endplates. Therefore,since there is no change in stagger between the two models, the flowstructures have very similar patterns except for the scale, which is afunction of the gap.

FIG. 106 compares the contours of the spanwise component of velocity (w)for the same configurations. The scale of the figure is from −1.5 m/s to1.5 m/s with 15 scales of resolution.

The spanwise component of velocity for Model #1 with a 1.0 C gap and nostagger at angles of attack 0°, 5° and 8° can be seen in FIG. 106( a).In the same manner, the figure for the 0.5 C gap and zero stagger areplotted in FIG. 106( b). This velocity component was found to have alarge variation with respect to angle of attack. As the angle of attackis increased, the magnitude of the horizontal velocity on the uppersurface gradually increases for both models. As with the horizontalvelocity component, there is a very similar pattern of flow structurebetween the two models.

FIG. 107 shows the vorticity for the two configurations at angles ofattack of 0°, 5° and 8° and a Re of 60,000. The scale of the figure isfrom (−) 200 rad/s to (+) 200 rad/s with 15 scales of resolution.

FIG. 108( a) represents the force balance measurements explained in theearlier section. The lift coefficient was obtained at and angle ofattack of 8° from the two different gap configurations seen above. Asdepicted in the figure, the 1.0 C gap configuration producesapproximately 27% higher CL than 0.5 C gap configuration. FIG. 108( b)represents the force balance measurements for the same configuration.These two curves show very similar behavior.

FIG. 109 shows the comparison of the vertical velocity components at thetwo different gap configurations for the upper and lower horizontalslices at an angle of attack of 8° and a Re of 60,000. The graph showsvery similar characteristics across the configurations. Although therewas a small variation between the curves on the upper line, gap spacingdid not alter the flow structure as profoundly as stagger.

FIG. 110 compares the magnitude of the vorticity at an angle of attackof 8° at the horizontal and vertical lines across the vortex core. Thisfigure shows similar trends in the magnitude of the vorticity as seen inthe vertical and spanwise velocity components from both the horizontaland vertical vortex core lines. These two unstagger configurations, havesimilar flow characteristics across the two gap spacings investigatedwith the magnitude of the vorticity increasing with increasing gap.

FIG. 111 shows the vortex core position relative to the wingtip andendplate for the two configurations at angles of attack of 5° and 8° anda Re of 60,000. The circular symbol in the figure represents the vortexcore position at α=5° and the triangular shape represents α=8°. As angleof attack increases there were small changes in the vortex corelocation, but the magnitude of the vorticity was significantly changed.Vortices occurred along the endplates for both models and the vortexcore moves up a little as gap increases. This roll-up behavior producesvortices and therefore, this causes the higher amount of downwash, whichwas displayed earlier. Therefore, there is almost no variation in thepatterns of the wingtip vortex between two different gaps with the samestagger, but the magnitude of the vorticity was different based on thesize of the gap. The endplates play an important role to pump theairflow toward the upper side and hence, if the gap is bigger, the sizeof the flow contour along the endplates grows. Since the resultingmomentum flux, which is generated by the upwash must be balanced; thedownwash also increases as the gap of the endplates increases. The forcebalance measurements and the downwash angle measured support thisargument.

Lift Force Calculation from Stereo PIV using Circulation Theory

Computation of the lift force is an important characteristic that canvalidate the Stereo PIV data generated during testing as well ascorrelate the integrated force results. To measure the lift force on themodels at Re 60,000 with angles of attack of 0°, 5° and 8° degrees,velocity field data in the Trefftz plane were obtained using the StereoPIV method. As discussed herein from circulation theory, lift force is afunction of density, velocity and circulation.

(L = ρ_(∞)V_(∞)∫_(c)Γ_(z) z).

Circulation was calculated using Stokes' theorem (equation 2.11,

Γ_(z) = ∫∫_(S)ω_(z)⋅ s = ∫∫_(S)(∇×V)⋅ s),

where ω is the vorticity,

${\omega_{z} = {{\nabla{\times V}} = \left( {\frac{\partial v}{\partial z} - \frac{\partial w}{\partial y}} \right)}},$

extracted from the DPIV results and ds refers to the area ofintegration.

Two sets of models were used to compute and compare lift coefficients asshown in Table below. Four models consist of varying stagger of (+) 1.0C, 0 C, (−) 0.5 C and (−) 1.0 C stagger at a constant 0.5 C gap acrossall models. FIG. 112 shows the comparison of the integrated forcemeasurement to the PIV derived circulation lift for the fourconfigurations.

The difference between the coefficients of lift obtained using a forcebalance and circulation theory is displayed in Table below. A potentialsource of error can be seen with the location of light sheet. For all ofconfigurations, the light sheet was located 0.82 chord lengthsdownstream from the longitudinal location in the middle between theupper and lower trailing edge. It was observed that this distance, 0.82chord lengths, was long enough to get fully developed wing-tip vorticesbecause the comparison of lift coefficients for the Model #2 (nostagger, 0.5 C gap) agrees well. However, for Model #3 ((−) 0.5 Cstagger, 0.5 C gap) and #4 ((−) 1.0 C stagger, 0.5 C gap), thecoefficients of lift that were obtained from circulation theory have an18.8% difference when compared to the force balance measurements at theobserved angles of attack of 5° and 10°.

Table showing difference between the coefficients of lift determined byintegrated force and circulation theory. (−) 0.5 C AoA (−) 1.0 C staggerstagger No stagger (+) 1.0 C stagger 5° 21.3% 19.0% 6.0% 10.6% 10° 18.7%  16% 4.4% 8.3%

For the two negative stagger configurations, the light sheet was located0.54 C and 0.3 C from the trailing edge of the upper wing respectivelyand therefore (see FIG. 113), this distance is likely not long enoughfor the wingtip vortex to roll up. This argument is supported by thefact that the lift coefficient obtained from circulation theory is lowerthan the force balance derived lift at the observed angles of attack.

FIG. 114 shows the integrated lift force for the 1.0 C and 0.5 C gapmodels without stagger. For both configurations, the light sheet waslocated 0.82 chord lengths downstream from the longitudinal location inthe middle between the upper and lower trailing edge. The coefficient oflift that was obtained from circulation theory closely agrees with theforce balance measurements at the observed angles of attack. This likelymeans that the wingtip vortex was completely rolled-up at the locationof the light sheet.

The difference between the coefficients of lift obtained using a forcebalance and circulation theory is displayed in Table below. For bothconfigurations, the agreement was higher than 94.3% on average at a Reof 60,000 and angles of attack 5° and 8°.

Table showing the difference between the coefficients of lift determinedby force balance and circulation theory. AoA 1.0 C Gap 0.5 C Gap 5° 3.9%6.0% 10° 8.5% 4.4%

This PIV analysis shows how the wingtip vortex is formed at thedifferent stagger and gap configurations of the biplane with endplates.The stagger effect shows a large variation with respect to the velocitycomponents and the wingtip vortex structure. The negative staggerconfigurations show several vortices separated and spread out at thelower wingtip and behind the upper wing. When the upper wing ispositively staggered, the biplane generates well-formed vortices. Thiscreates an upwash outboard of the endplates as well as a vortex at thetrailing edge of the wing and endplates.

This upwash produces the same amount of downwash from both upper andlower wing surfaces by the momentum theory of lift since the resultingmomentum flux by upwash must be balanced assuming no change in the outof plane motion. From the observation of the vortex core location, thepositive stagger configuration produces higher downwash and therefore,the lift force obtained was higher than that obtained with the otherconfigurations. The combination of force balance results for lift andthe downwash angle prove this flow behavior. For the gap effect, as gapincreases, this wingtip vortex roll-up behavior was similar for both the0.5 C and 1.0 C gap models but the magnitude of the vorticity for thelarger gap configuration was higher. Therefore, as the gap increases fora given stagger condition a higher lift coefficient is obtained. Theeffect of gap did not change the vortex morphology as much as the changein stagger when viewed at the same location.

It will be appreciated that this invention has application to a numberof significantly different situations where fluid drag is encountered;for example to fixed wing aircraft, rotary wing aircraft, submarines andhydrofoils. Those skilled in the art will understand that manymodifications may be made to structures, materials, proportions,arrangements, components and methods described herein, without departingfrom the scope of the invention claimed below.

1. In a lifting foil the combination of (a) a lower trailing coursehaving a lower starboard: margin and a lower port margin, said lowercourse extending sideward between said lower starboard margin and saidlower port margin and being responsive to fluid flow in a direction −X,for generating a first fluid reaction force having a first liftingcomponent in a direction +Z, (b) an upper leading course having an upperstarboard margin and an upper port margin, said upper course beingpositioned above said lower trailing course and extending sidewardbetween said upper starboard margin and said upper port margin forgenerating a second fluid reaction force having a second liftingcomponent in said direction, +Z, parallel and additive to said firstreaction force, (c) a starboard flow guide extending vertically betweensaid lower starboard margin and said upper starboard margin, having aprogressively adjusted camber for suppressing spanwise fluid flow in adirection +Y, perpendicular to X and Z, and (d) a port flow guideextending vertically between said lower port margin and said upper portmargin, having a progressively adjusted camber for suppressing spanwisefluid flow in a direction −Y, opposite to +Y.
 2. A lifting foilaccording to claim 1, wherein said flow guides are secured to saidcourses by smooth, continuous connections which blend into said coursesat said margins.
 3. A lifting foil according to claim 1 wherein saidlifting foil has a vertical cross-section which is generally elliptical.4. A lifting foil according to claim 1 wherein said fuselage is situatedmidway between said flow guides with a cargo compartment between saidmargins.
 5. A lifting foil according to claim 1 wherein said fuselage issituated midway between said flow guides, said lifting foil beingprovided with a large starboard passage situated between said fuselageand said starboard flow guide, and a large port passage situated betweensaid fuselage and said port flow guide.
 6. A lifting foil according toclaim 1 further having three surfaces comprising a first surfacedefining said port passage, a second surface defining said starboardpassage and a third surface defining the exterior shape of said liftingfoil.
 7. A lifting foil according to claim 6, said fuselage beingenclosed by cooperative coverage from said first, second and thirdsurfaces.
 9. A lifting foil comprising: (a) A lower trailing coursehaving a lower starboard margin and a lower port margin, said lowercourse extending sideward between said lower starboard margin and saidlower port margin for generating a first fluid reaction force having anupwardly directed lifting component, (b) An upper leading course havingan upper starboard margin and an upper port margin, said upper coursebeing positioned above said lower course and extending sideward betweensaid upper starboard margin and said upper port margin for generating asecond fluid reaction force having an upwardly directed liftingcomponent, (c) A starboard flow guide extending vertically between saidlower starboard margin and said upper starboard margin, said starboardflow guide comprising: (i) a first upper section secured to said upperstarboard margin, (ii) a first lower section secured to said lowerstarboard margin, and (iv) a first mid-section positioned between saidfirst upper section and said first lower section, (d) A port flow guideextending vertically between said lower poll margin and said upper portmargin, said port flow guide comprising: (i) a second upper sectionsecured to said upper port margin, (ii) a second lower section severedto said lower port margin, and (iv) a second mid-section positionedbetween said second upper section and said second lower section, saidstarboard flow guide and said port flow guide having camberedcross-sections characterized by a camber which is substantially zero atan associated flow guide mid-point and which increases progressively tomaximum values at opposite ends of associated flow guides.
 10. A liftingfoil comprising: (a) an upper leading course including an upperstarboard end and an upper port end, (b) a lower trailing courseincluding a lower starboard end and a lower port end, said lower coursebeing generally parallel to said upper course with said lower starboardend facing said upper starboard end and said lower port end facing saidupper port end, (c) a starboard flow guide having an upper starboardguide section connected to said upper starboard end, a lower starboardguide section connected to said lower starboard end, and a starboardmid-section connecting said upper starboard guide section to said lowerstarboard guide section, and (d) a port flow guide having an upper portguide section joined to said upper port end, a lower port guide sectionjoined to said lower port end and a port mid-section connecting saidupper port guide section to said lower port guide section; saidstarboard flow guide having a camber which increases progressively froma minimum camber at a mid-point of said starboard flow guide to amaximum camber at said upper starboard end and also being camberedprogressively from said mid-point to said lower starboard end said portflow guide having a camber which increases progressively from a minimumcamber at a mid-point of said port flow guide to a maximum camber atsaid upper port end and also being cambered progressively from saidmid-point to said lower port end.